Question

The first term of a progression is ( sin ^{2} heta ), where ( 0< heta<frac{1}{2} pi ). The second term of the progression is ( sin ^{2} heta cos ^{2} heta ). (a) Given that the progression is geometric, find the sum to infinity. [3] It is now given instead that the progression is arithmetic. (b) (i) Find the common difference of the progression in terms of ( sin heta ). [3] (ii) Find the sum of the first 16 terms when ( heta=frac{1}{3} pi ). [3]

          The first term of a progression is ( sin ^{2} 	heta ), where ( 0<	heta<frac{1}{2} pi ). The second term of the progression is ( sin ^{2} 	heta cos ^{2} 	heta ).
(a) Given that the progression is geometric, find the sum to infinity.
[3]

It is now given instead that the progression is arithmetic.
(b) (i) Find the common difference of the progression in terms of ( sin 	heta ).
[3]
(ii) Find the sum of the first 16 terms when ( 	heta=frac{1}{3} pi ).
[3]

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$The first term of a progression is ( sin ^2 heta ), where ( 0< heta<frac12 pi ). The second term of the progression is ( sin ^2 heta cos ^2 heta ). (a) Given that the progression is geometric, find the sum to infinity. [3] It is now given instead that the progression is arithmetic. (b) (i) Find the common difference of the progression in terms of ( sin heta ). [3] (ii) Find the sum of the first 16 terms when ( heta=frac13 pi ). [3]$

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Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

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Solved by Expert Eduard Sanchez

02/05/2025

Step 1

The first term is \( a_1 = \sin^2 \theta \) and the second term is \( a_2 = \sin^2 \theta \cos^2 \theta \). Show more…

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Transcript

0:00 Hi there.

00:01 So for this problem, for part a of this problem, we are going to use the geometric progression case.

00:13 For this, the first term will be 8 that is equal to the sine square of theta, and then the terms 2 will be the sine square of theta times the cosine square of theta.

00:28 Since this is a geometric progression, the common ratio will be the term 2 divided by the term 1.

00:36 So that will be then the sign square of theta times the cosine square of theta.

00:41 This divided by the sign square of theta.

00:45 This will give us the cosine square of theta.

00:49 The sum to infinity of a geometric progression is given by the following expression.

00:55 That will be 8 divided by 1 minus 0.

00:58 So then in this case, that will be the sine square of theta, this divided by one minus the cosine square of theta.

01:07 Simplifying this, this will give us just simply, sine square of theta divided by the sign square of theta.

01:15 That will give us one.

01:17 And that will be the sum to infinity for part a of this problem.

01:23 Now for part b, we are asked about the argument deprivation case.

01:26 For the first case of this, we are asked to find the common difference in terms of the sign of theta.

01:32 So then we know this because the difference will be the term two minus the term one.

01:37 So from this, we will have to, that is the sign square of theta times the cosine square of theta minus the sign square of theta...

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