WEBVTT

43d231d7-3782-4aa6-a656-5a0aee89c2f8-0
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Hi I'm Fiona and this is Tamua 2021 paper
one and question three.

ec8f008e-a703-4c3f-8b44-2e3ddf662f67-0
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We are told that an arithmetic
progression and a convergent geometric

ec8f008e-a703-4c3f-8b44-2e3ddf662f67-1
00:00:16.028 --> 00:00:18.200
progression each have first term 1/2.

0a2d25ad-5123-4069-97f5-6bf02fb143d3-0
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The sum of the second terms of the two
progressions is 1/2 and the sum of the

0a2d25ad-5123-4069-97f5-6bf02fb143d3-1
00:00:23.579 --> 00:00:26.320
third terms of the two progressions is
1/8.

c6a452a2-7f30-4fba-a345-c8bca3745c03-0
00:00:26.760 --> 00:00:32.360
And we are asked to find the sum to
Infinity of the geometric progression.

b76b88e9-c45a-4732-8439-92dbb8ba26e8-0
00:00:33.160 --> 00:00:37.760
We're given options A to H to choose from
when we get to the end of the question.

b9d29393-963e-4068-8a5f-f8ecef19d681-0
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And 1st off,
I just want to clarify that a progression

b9d29393-963e-4068-8a5f-f8ecef19d681-1
00:00:42.254 --> 00:00:44.520
is the same thing as a sequence.

e5890425-233c-4be0-a190-a4ca6b075105-0
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So this convergent geometric progression
and arithmetic progression that we're

e5890425-233c-4be0-a190-a4ca6b075105-1
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going to be looking at in this question,
and they are sequences.

99376249-1fe4-4d9f-9267-b6a895273145-0
00:00:55.240 --> 00:00:57.689
So we're told something about the first
term,

99376249-1fe4-4d9f-9267-b6a895273145-1
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and we're also it looks like we're going
to need information about the second

99376249-1fe4-4d9f-9267-b6a895273145-2
00:01:01.843 --> 00:01:04.080
terms of both and the third terms of both.

188d176b-02e3-47c4-b89c-09181250656b-0
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So I'm going to draw myself a little
table to help collect that information.

c0bc9d69-7936-488e-8dee-75cb1433e4b5-0
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OK, here's my table.

31df9131-ece3-41d5-af38-4090d4040562-0
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Let me start populating it.

83d992d4-e548-4892-9201-b381ef529f1d-0
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So I'm told that the first term of both
is 1/2 for an arithmetic progression.

8c71cf17-609c-4b3d-a7b0-7f2e85bfbdef-0
00:01:29.860 --> 00:01:32.100
I know that the second term is.

9c2bf790-da4c-4e5c-b33e-6d467834d569-0
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We get the second term by taking the
first term and adding the common

9c2bf790-da4c-4e5c-b33e-6d467834d569-1
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difference on to it.

b0dee8fa-40c7-45bf-9c3c-bd71db400a30-0
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And we usually label the common
difference D So the second term of this

b0dee8fa-40c7-45bf-9c3c-bd71db400a30-1
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arithmetic progression is going to be 1/2
+ d and the third term is going to be

b0dee8fa-40c7-45bf-9c3c-bd71db400a30-2
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when we add another D on.

ee177a76-d939-4057-a2ab-b30c6b7a0fb3-0
00:01:50.040 --> 00:01:56.159
So the third term is going to be 1/2 + 2
D For the geometric progression,

ee177a76-d939-4057-a2ab-b30c6b7a0fb3-1
00:01:56.159 --> 00:01:58.640
the first term we know is 1/2.

6c3040ff-5314-4727-b4c8-4d13eb1c4496-0
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The second term we find by multiplying
the first term by the common ratio,

6c3040ff-5314-4727-b4c8-4d13eb1c4496-1
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which we usually label R.

b1505b16-b1eb-490a-9700-f10cf2d19822-0
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And so the second term for this geometric
progression will be 1/2 * r.

73e174af-33f3-45f6-9a38-445c6c29f75d-0
00:02:10.960 --> 00:02:16.989
And the third term will be when we
multiply by another R So the third term

73e174af-33f3-45f6-9a38-445c6c29f75d-1
00:02:16.989 --> 00:02:19.080
is going to be 1/2 * r ^2.

89a8b7c0-d86e-4390-8413-bc53f67d0d29-0
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Now I'm going to be able to use the
information in these two lines.

54925aff-5e35-42d3-9540-a3e619cb6343-0
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So the first line is tells me that the
sum of the second terms of the two

54925aff-5e35-42d3-9540-a3e619cb6343-1
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progressions is 1/2.

183f2ba1-d9d1-4365-a5f4-de63fdc8fef1-0
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And using the information in my table,
I have 1/2 + d + 1/2 * r,

183f2ba1-d9d1-4365-a5f4-de63fdc8fef1-1
00:02:37.114 --> 00:02:41.640
which is r / 2 and that's going to equal
1/2.

17d1d402-080a-4b92-8285-1ee7591b8c96-0
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OK,
so that tells me some information about

17d1d402-080a-4b92-8285-1ee7591b8c96-1
00:02:44.898 --> 00:02:45.440
D&R.

18e0915d-d27b-4729-b0f3-54cac4e1d825-0
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The next line tells me that the sum of
the third terms of the two progressions

18e0915d-d27b-4729-b0f3-54cac4e1d825-1
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is 1/8.

ba53b38e-79bb-49f0-9e91-ebc0a881aa59-0
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So that gives me 1/2 + 2 D plus 1/2 * r
^2 which is r ^2 / 2 is going to equal

ba53b38e-79bb-49f0-9e91-ebc0a881aa59-1
00:03:01.940 --> 00:03:02.480
1/8.

5a37d5d5-3754-456f-a56d-e31ce5574d13-0
00:03:03.480 --> 00:03:09.235
Now I need to find the sum to Infinity of
this geometric progression and there is a

5a37d5d5-3754-456f-a56d-e31ce5574d13-1
00:03:09.235 --> 00:03:10.400
formula for that.

6ddd9ee3-c0af-47cf-896f-cb250b0c9c7a-0
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It is South Infinity,
which is the notation that we use for the

6ddd9ee3-c0af-47cf-896f-cb250b0c9c7a-1
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sum to Infinity.

258aaf46-4a4d-4db7-9d45-106109daa472-0
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And it's going to be A which is the first
term over 1 -,

258aaf46-4a4d-4db7-9d45-106109daa472-1
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r So that means I'm going to need to find
RI,

258aaf46-4a4d-4db7-9d45-106109daa472-2
00:03:25.074 --> 00:03:28.919
already know A because I was told a in
the question.

d6f4901b-a4ec-4e5a-b5ba-43daf829d481-0
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It's a half,
but I'm going to need to find R and I'm

d6f4901b-a4ec-4e5a-b5ba-43daf829d481-1
00:03:33.006 --> 00:03:38.634
going to use these two pieces of
information to find R I can see that in

d6f4901b-a4ec-4e5a-b5ba-43daf829d481-2
00:03:38.634 --> 00:03:44.571
this equation I have an R-squared,
so I'm going to rearrange this one to get

d6f4901b-a4ec-4e5a-b5ba-43daf829d481-3
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D in terms in terms of R substitute that
in here,

d6f4901b-a4ec-4e5a-b5ba-43daf829d481-4
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and then I have an equation in R So
multiplying all across by two,

d6f4901b-a4ec-4e5a-b5ba-43daf829d481-5
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I get 1 + 2 D plus r = 1.

1d787051-3e65-4258-ad25-c1d7f34a6a1c-0
00:03:55.960 --> 00:04:09.341
So that means 2D plus r = 0 and that
means that two d = - r and that d = - r /

1d787051-3e65-4258-ad25-c1d7f34a6a1c-1
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2.

009458b9-1579-4bf5-9517-f9b2ad61b061-0
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OK substituting this value of D into this
equation here gives me 1/2 plus.

0da5a782-feea-4d85-ae1a-836f7a906b70-0
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If D is minus r / 2,
then this term is going to be -r + r ^2 /

0da5a782-feea-4d85-ae1a-836f7a906b70-1
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2 = 1/8.

16396699-f643-4d35-b9ba-bfa36932d9f8-0
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OK, I'm going to multiply all across by 8.

121a02f6-899b-48a7-8117-b453687511a7-0
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So I get 4 -, 8 R Plus four r ^2 = 1.

bcb617dc-8a87-43e3-b64a-1483794b3ea3-0
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And then rearranging this quadratic I
have 4 R-squared -8 R and then here's a

bcb617dc-8a87-43e3-b64a-1483794b3ea3-1
00:04:52.660 --> 00:04:59.000
+4 take away one which is going to leave
me with a positive 3 = 0.

17f59a06-33ef-40cc-a94b-c6254fe5541f-0
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So now all I need to do is factorise this
quadratic.

70217696-8b7d-40f4-bfa0-54c4b337cab2-0
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I'm hoping it will give me a nice
factorization.

fce081aa-d6d2-4da2-b1e9-23dd4024a309-0
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Usually it does with Tamila.

fd09887f-0ef4-41ea-aa64-b7b89cc27130-0
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So let's see if I do 2R here and two R
here, then this 3 can be a - 3 * a - 1.

e76ea662-b731-4ee4-bff7-cc088e728b91-0
00:05:17.040 --> 00:05:27.781
So it would give me -6 - 2 and that gives
me that -8 so -3 - 1 and that all equals

e76ea662-b731-4ee4-bff7-cc088e728b91-1
00:05:27.781 --> 00:05:28.040
0.

45919d93-a013-40eb-af3a-70bbc4ff0da9-0
00:05:28.560 --> 00:05:36.200
So this factor here is going to give me a
value of R of r = 3 / 2.

9d5c9bcb-e61c-46d0-873c-38f82bf2608c-0
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This one is going to give me a value for
R of r = 1/2.

b4f0cdaf-74ce-4682-a4a7-6b466c4fb9c3-0
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And now I go back to the information I'm
given in the question which told me that

b4f0cdaf-74ce-4682-a4a7-6b466c4fb9c3-1
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the geometric progression is convergent.

c8e9c53e-0c80-44fd-96fb-810d1f31503c-0
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Now in order for a geometric progression
or a geometric sequence to be convergent,

c8e9c53e-0c80-44fd-96fb-810d1f31503c-1
00:05:56.493 --> 00:05:59.493
I need the modulus of R to be less than
one,

c8e9c53e-0c80-44fd-96fb-810d1f31503c-2
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which is another way of saying I need R
to be between -1 and 1.

cd4b7649-0620-4144-a201-3a5733926744-0
00:06:04.160 --> 00:06:11.960
And so the value of R which is feasible
is r = 1/2.

bbd9e59e-c1ab-46b5-8c87-2e511ff7b3e9-0
00:06:12.400 --> 00:06:20.699
That means that for S Infinity I'm
looking at 1/2 which is my first term,

bbd9e59e-c1ab-46b5-8c87-2e511ff7b3e9-1
00:06:20.699 --> 00:06:24.400
which is a / 1 -, r which is 1/2.

d65ec39c-9bac-4422-b480-2b6bcc79d7b5-0
00:06:24.800 --> 00:06:35.000
So 1/2 / 1 - 1/2 is 1/2 / 1/2 which is
going to be 1/2 / 1/2 which is 1.

3324fa36-68bd-4cba-9ddc-7796c4bc23f7-0
00:06:36.200 --> 00:06:40.958
So looking over at my options that I have
to choose from that gives me option G

3324fa36-68bd-4cba-9ddc-7796c4bc23f7-1
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which is the answer to this question.

02e34a69-6afc-4476-b792-85e70035eda6-0
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Let's take some time to reflect on this
question.

65a9842a-2841-4f56-8818-c517c62915eb-0
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Firstly,
writing out information in the form of a

65a9842a-2841-4f56-8818-c517c62915eb-1
00:06:51.010 --> 00:06:53.360
table can be really valuable.

315f4400-a0db-49ab-a29c-4b4573565881-0
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This question gave me information about
the first term and told me ways in which

315f4400-a0db-49ab-a29c-4b4573565881-1
00:06:58.733 --> 00:07:02.040
I needed to then use the second terms and
3rd terms.

5255ddab-9c59-4cb1-b9be-3aad4ff5179a-0
00:07:02.320 --> 00:07:07.347
And so drawing out this table allowed me
to compile that information and then

5255ddab-9c59-4cb1-b9be-3aad4ff5179a-1
00:07:07.347 --> 00:07:11.537
track it as I went along,
which was really valuable to answering

5255ddab-9c59-4cb1-b9be-3aad4ff5179a-2
00:07:11.537 --> 00:07:12.440
this question.

85b88c38-179e-4eb9-9257-05c0e48e1862-0
00:07:13.240 --> 00:07:17.958
Second thing I want to reflect on is this
geometric progression that we were

85b88c38-179e-4eb9-9257-05c0e48e1862-1
00:07:17.958 --> 00:07:22.738
looking at which had the first term 1/2
that was given in the question and we

85b88c38-179e-4eb9-9257-05c0e48e1862-2
00:07:22.738 --> 00:07:25.680
ended up finding that the common ratio
was 1/2.

12022b26-16c1-4a94-a208-2ab6941fb5f3-0
00:07:25.680 --> 00:07:30.640
And so that means the second term would
be 1/2 * 1/2, which is 1/4.

44a26ecc-0fc2-4e04-8fc2-fe32bb8897c4-0
00:07:30.960 --> 00:07:34.840
And then we would multiply by 1/2 again
to get the third term, which would be 1/8.

dcc6da14-6b49-40bf-bef2-896f7382494e-0
00:07:35.160 --> 00:07:40.723
And we would multiply it by 1/2 again to
get the fourth term which would be 116th

dcc6da14-6b49-40bf-bef2-896f7382494e-1
00:07:40.723 --> 00:07:42.080
and so on and so on.

c2501ce6-581e-4aaf-8ff8-c9666673fea8-0
00:07:42.960 --> 00:07:49.836
Now the nice way to visualise the sum to
Infinity of this geometric progression

c2501ce6-581e-4aaf-8ff8-c9666673fea8-1
00:07:49.836 --> 00:07:53.360
with first term 1/2 and common ratio 1/2.

7ba7e6ee-5918-4ce8-8de8-da4d5eb1542f-0
00:07:53.880 --> 00:07:58.436
If I imagine that this bar here takes a
value of 1,

7ba7e6ee-5918-4ce8-8de8-da4d5eb1542f-1
00:07:58.436 --> 00:08:05.358
then let's just go through and add each
term of this geometric progression and

7ba7e6ee-5918-4ce8-8de8-da4d5eb1542f-2
00:08:05.358 --> 00:08:08.600
see how it approaches the value of 1.

48ef8e56-626b-4da1-953b-1ac83d0791cf-0
00:08:08.880 --> 00:08:14.960
So the first term has value of half,
which I can add on here.

606ccd70-b440-436a-b817-33552e943b76-0
00:08:15.400 --> 00:08:19.666
The second term has value 1/4,
which I can add on here and visualise in

606ccd70-b440-436a-b817-33552e943b76-1
00:08:19.666 --> 00:08:20.200
this way.

4724bfc8-1299-40f4-bc12-551618932e76-0
00:08:20.520 --> 00:08:26.637
The third term has value 1/8,
the fourth term has a value 116th,

4724bfc8-1299-40f4-bc12-551618932e76-1
00:08:26.637 --> 00:08:28.520
and so on and so on.

11027423-59d6-4b13-ae60-6bba5012375d-0
00:08:28.520 --> 00:08:34.240
I'd be adding 132 here, 164th here,
etcetera, etcetera.

e859519c-3fd0-42e0-a726-3bf1c56b371e-0
00:08:34.240 --> 00:08:42.816
And can you see how the sum of the terms
of this geometric progression converge to

e859519c-3fd0-42e0-a726-3bf1c56b371e-1
00:08:42.816 --> 00:08:45.400
or approach a value of 1?

389c9353-a414-4135-9853-fd41b414e51d-0
00:08:45.640 --> 00:08:48.337
So if you've seen something like this
before,

389c9353-a414-4135-9853-fd41b414e51d-1
00:08:48.337 --> 00:08:52.500
then if you're in an exam setting and
you're under some time pressure,

389c9353-a414-4135-9853-fd41b414e51d-2
00:08:52.500 --> 00:08:57.016
that might just give you a little edge
when you're looking at the options to

389c9353-a414-4135-9853-fd41b414e51d-3
00:08:57.016 --> 00:08:57.719
choose from.

513ab648-0af4-4eee-b186-12b7c3d59051-0
00:08:57.920 --> 00:09:03.108
And you might be able to more quickly see
that this geometric progression or the

513ab648-0af4-4eee-b186-12b7c3d59051-1
00:09:03.108 --> 00:09:06.760
sum to Infinity of the geometric
progression would be 1.