Question

The sum of the first two terms of two series, one
in A.P. and the other in G.P. is the same. If the first term
of both series is 2/3, and the common difference of the
A.P. and the common ratio of the G.P is equal. Find the
sum of first 20 terms of the AP.
(a) - 1100/3
(b) 1100/3
(c) 1102/3
(d) - 1150/3.
​

Answer 1

Answer:

Option A is correct.

Step-by-step explanation:

As it is given that

AP: 2/3,2/3+d,...

GP= 2/3,2r/3,...

It is given that

d= r

and

$\frac{2}{3} + \frac{2}{3} + d = \frac{2}{3} + \frac{2r}{3} \\ \\ \frac{2 + 2 + 3d}{3} = \frac{2 + 2r}{3} \\ \\ 4 + 3d = 2 + 2r \\ \\ since \: r = d \\ \\ 4 + 3d = 2 + 2d \\ \\ 3d - 2d = 2 - 4 \\ \\ d = - 2 \\ \\ hence \: r = - 2$

Sum of 20 terms of AP is

$S_n = \frac{n}{2} \bigg(2a + (n - 1)d\bigg) \\ \\ S_{20} = \frac{20}{2}\bigg (2 \times \frac{2}{3} + (20 - 1)( - 2)\bigg) \\ \\ = 10\bigg( \frac{4}{3} - 38\bigg) \\ \\ = 10\bigg( \frac{4 - 114}{3} \bigg) \\ \\ = 10\bigg( \frac{ - 110}{3} \bigg) \\ \\ S_{20} =- \frac{1110}{3}$

Hope it helps you.

Answer 2

Answer:

the above information is correct