WEBVTT

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Hi I'm Fiona and this is Tamila 2021
paper one and question 13.

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We are told that the function F is such
that for every integer N,

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the integral between N and n + 1 of F of
X with respect to X = n + 1.

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And we're asked to evaluate the sum from
r = 1 to r = 8 of of the integral between

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zero and R of F of X with respect to X.

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And then we are given and then we are
given options to go through once we get

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to a conclusion.

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Now first of all,
I just want to get a feel for what I'm

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being asked to sum.

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So I'm going to draw a table to help me.

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So here's my table.

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I'm going to go through and consider each
value of R and what value the integral

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gives for each value of R So when r = 1,
this integration will be the integral of

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F of X with respect to X between zero and
one.

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This result up here tells me that if I'm
integrating between two consecutive

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integers,
then the value of the integral will be

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the largest integer.

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So when r = 1,
I'm integrating between zero and one,

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so the value of the integral will be 1.

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When r = 2,
I've got I'm integrating between zero and

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two now.

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But I can think of the integral between
zero and two being the same as the

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integral between zero and one plus the
integral between 1:00 and 2:00 because

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I'm just adding areas.

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And so here the value of the integral for
r = 2 will actually be two plus one.

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So that's going to be 3.

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When r = 3,
I'm going to be considering the integral

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between zero and one,
which will be a value of 1.

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Then the integral between 1:00 and 2:00,
which will be a value of two,

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and then the integral adding on the
integral between 2:00 and 3:00,

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which will give me a value of three.

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So my value,
I'll get my value by adding 1 + 2 + 3,

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which is 6.

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And now I'm going to fill in the rest of
the table.

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Now the next thing I need to do is to
just add these values together,

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and that will give me the value of the
summation that I'm asked to evaluate.

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When I Add all of these together together,
I get a value of 120.

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And looking at my options,
I can see that that's option C.

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So option C is the answer to this
question.

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Let's take some time to reflect on this
question.

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When I generated each of these values
here,

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I did that by adding consecutive integers
from one onwards.

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So for example, to get this value 10,
I added 1 + 2 + 3 + 4.

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And when we generate numbers in this way,
they're called triangle numbers.

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There is actually a formula for the sum
of triangle numbers which is the sum that

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I had to do here.

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Here I was adding all the triangle
numbers up to the 8th so I can use the

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formula for the sum of the first N
triangle numbers which is N times north

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plus one times north +2 all over 6.

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Here I'm adding the 1st 8 triangle
numbers.

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So my sum would or my value that I'm
looking for it would be 8 * 9 * 10 all

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over 6/6 is 2 * 3.

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So I'm going to take a factor of 2 from
the 8 here and a factor of three from the

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9.

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And I can see there that my answer will
be 4 * 3 which is 12 * 10 which is 120.

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And that is possibly a more efficient way
to answer this question if you spot that

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really you're summing the 1st 8 triangle
numbers.