Question

Question 7 If f is a function satisfying f(x+y) = f(x)(y) for all x, y ϵ N such that f(1) = 3 and f(x) = 120 , find the value of n.

Class X1 - Maths -Sequences and Series Page 199

Answer 1

Given,
f(x + y) = f( x ). f( y )
if we put x = y = 1
f( 1 + 1) = f( 1 ) .f( 1 )
f(2) = 3 . 3
f(2) = 9

putting , x = 2 and y = 1
f( 2 + 1) = f(2).f(1)
f(3) = 9 × 3
f(3) = 27

putting , x = 3 , y = 1
f(3 + 1) = f(3).f(1)
f(4) = f(3) .f(1)
f(4) = 27 × 3
f(4) = 81

now, A/C to question,
∑^{n}_{x=1} f(x) = 120
f(1) + f(2) + f(3) + ......... + f(n) = 120
3 + 9 + 27 + .......+ n terms = 120
we see series is in GP
first term ( a) = 3
common ratio ( r) = 3
then, Sn = a( rⁿ - 1)/(r - 1) use this formula here,

3(3ⁿ - 1)/(3 - 1) = 120
(3ⁿ - 1) = 40 × 2
3ⁿ - 1 = 80
3ⁿ = 81
3ⁿ = 3⁴
n = 4