Question

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

Answer 1

Explanation:

Let the 3 number in the geometric sequence be a/r, a and ar.

The product of the three numbers is a^3 = 512, So a = 8.

Now (a/r) +8, a+6, ar form an AP.

So we have: ar - (a+6) = a+6 - (a/r)-8, or

ar^2-ar-6r = ar+6r-a-8r, or

8r^2–8r-6r = 8r+6r-8–8r, or

8r^2–14r-6r +8 = 0

8r^2–20r+8 = 0, or

2r^2–5r +2= 0, or  

(2r-1)(r-2) = 0

Hence r =2 or 1/2.  

So the three terms of the GP are 4, 8 and 16 or 16, 8 and 4.

Check: 4+8, 8+6, 16 = 12, 14 16, all in AP. Correct.  

16+8, 8+6, 4 = 24, 14, 4, all in AP. Correct.