Question

If the 5th term of a G.P is 16 and the 10th term is 1/2,then the 15th term is

Answer 1

Answer:

given,

5th term of a G.P = 16.

10th therm of a G.P = 1/2.

to find :-

15th term in G.P.

$\huge\sf\underline{solution}$

let ,

the first term of the G.P = 'a'

common ratio = 'r'.

nth term of G.P = a_n = a×(r^n-1)

. : 5th term = a_5 = a×(r^5-1) = 16

16 = a × (r⁴) -------(1)

10th term of G.P = a_10 = a×(r^10-1) = 1/2

1/2 = a × (r⁹) ----(2)

(2)/(1)

1/2/16 = a×r⁹/a×r⁴

1/32 = r⁵

(1/2)⁵ = r⁵

.: r = 1/2

substituting value of r in eq 1:-

16 = a×(1/2)⁴

16 = a × 1/16

a = 16×16

a = 256.

a_15 = a × (r¹⁴)

= 256 × (1/2)¹⁴

= 256 × 1/16384

= 64.

.: a_15 = 64

Answer 2

Given information :

5th term of G. P = 16

10 th term of G. P =1/2

To Find :

15 th term

Solution :

nth term of G. P = a ( r^n-1 )

take n=5

=> a_5= a×( r^ 5-1 )= 16

=> a_5= a× r^4 =16

=> a×r^4=16 ------- equation- 1

take n=10

=> a_10 = a× (r^10-1) ==1/2

=> a_10= a× r^9 =1/2

=> a× r^9 = 1/2 ------- equation- 2

equation 2 ÷ equation 1

1/2 /16 = a×r^9 / a × r^ 4

1/32 = r^5

(1/2)^5 = r^5

.: r=1/2

Substituting value of r in equation 1

a×r^4=16

a×(1/2)^4 =16

a×1/16 =16

a=16/1/16

a=16×16

:.a=256

Now substitute a value in a_ 15

a_15 =a×(r^15-1)

a_15 = 256 × (1/2^14 )

a_15 =256 × 1/16384

a_15 = 64

Answer : 64