The first, second and third terms of an arithmetic...

The first, second and third terms of an arithmetic progression are ( a, frac{3}{2} a ) and ( b ) respectively, where ( a ) and ( b ) are positive constants. The first, second and third terms of a geometric progression are ( a, 18 ) and ( b+3 ) respectively. (a) Find the values of ( a ) and ( b ). [5] (b) Find the sum of the first 20 terms of the arithmetic progression. [3]

Transcript

0:00 Hi there.

00:01 So for this problem, let's start with part a of this problem, where we are asked to find the values of a and b.

00:08 So for this, we are going to use arignetted progression, the condition.

00:12 So the first three terms of the intermittent progression, and we are given, it is a, 3 divided by 2 times a and b.

00:22 The common difference between this, that will be that t2 minus t1 equals to 3, 3 3, 3, 3, 3, minus d2.

00:31 So substituting the values, we will have that 3 divided by 2 times a minus a should be equal to b minus 3 divided by 2 times a.

00:41 Simplifying this expression we will obtain, for simple solving for b, we'll obtain that b equals to just simply 2 times a.

00:51 So we have that then from this b equal to 2 times a.

00:55 Now we are going to use the geometric progression condition for this.

00:59 So the first three terms of a geometric progression are a, 18, and b plus 3.

01:07 So the common ratio will be that t2, the term 2, divided by the term 1, equals to the term 3, divided by the terms 2.

01:17 So in this case, we will have 18 divided by a equals to b plus 3 divided by 18.

01:25 Now we just solve in here, and solving in here we will obtain that 320.

01:31 24 equals to 2 times a square plus 3 times a.

01:36 When we substitute the condition in here, that b is 2 times a.

01:40 Now in here we will have a quadratic form...

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