Question

the product of the first five term of a geometric series is 243.if the third term of geometric series is equal to the tenth term of an arithmetic series.find the sum of the first 19 term of the arithmetic series​

Answer 1

Given:

• The product of the first five terms of a geometric series is 243
• The third term of the geometric series is equal to the tenth term of an arithmetic series

To find:

• The sum of the first 19 terms of the arithmetic series

We take both the geometric and arithmetic series to be progressions in order to solve the problem:

Let the five tems of the G. P. are

a/r^2, a/r, a, ar, ar^2

Given that, the product of these terms = 243

or, a/r^2 * a/r * a * ar * ar^2 = 243

or, a^5 = 243

or, a^5 = 3^5

or, a = 3

So, the third term of the G. P. is 3.

We consider the A. P. as

b, b + d, b + 2d, ..., b + (n - 1)d, ...

Here, the 10th term of the A. P. is

= b + (10 - 1) d

= b + 9d

ATQ, b + 9d = 3

∴ the sum of the first 19 terms of the A. P. is

= 19/2 * [2b + (19 - 1) d]

= 19/2 * (2b + 18d)

= 19 * (b + 9d)

= 19 * 3

= 57.

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Answer 2

Answer:

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Step-by-step explanation: