Question

Solve previous year Question of iit jee

Chapter :- sequence and series​

Answer 1

Question :-

The first two terms of a geometric progression add upto 12. The sum of the third and the fourth term is 48. If the terms of the geometric progression are alternately positive and negative, then the first term is ______ .

Solution :-

General series of a Geometric Progression :-

a, ar, ar², ar³, . . . . . . ., arⁿ⁻¹

where

a = First term of the Geometric Progression

r = Common ratio of the Geometric Progression

n = Order of term

It is given that,

The first two terms of a geometric progression add upto 12.

a + ar = 12     . . . . Equation (i)

The sum of the third and the fourth term is 48.

ar² + ar³ = 48 . . . Equation (ii)

Dividing both equations,

$\sf{\dfrac{a+ar}{ar^2+ar^3}=\dfrac{12}{48}}$

$\sf{\dfrac{a(1+r)}{ar^2(1+r)}=\dfrac{12(1)}{12(4)}}$

Canceling few terms,

$\sf{\dfrac{1}{r^2}=\dfrac{1}{4}}$

$\sf{\dfrac{1}{r}=\pm\dfrac{1}{2}}$

$\sf{r=\pm2}$

Since, the terms of this geometric progression are alternately positive and negative. Hence, 'r' will be negative.

Thus,

$\boxed{\sf{r = -2}}$

Substituting value of 'r' in equation (i),

a + ar = 12

a + a(-2) = 12

a - 2a = 12

-a = 12

a = -12

Answer :-

The first term of this geometric progression is -12.

Hence, option C is correct option.

Answer 2

Option (c) is the answer.