Home
Class 12
MATHS
Let f(x)=x+f(x-1) for AAx in R. If f(0)...

Let `f(x)=x+f(x-1)` for `AAx in R`. If `f(0)=1,f i n d \ f(100)`.

Text Solution

Verified by Experts

Given `f(x)=x+f(x-1) " and " f(0)=1`
Put `x=1`. Then,
`f(1)=1+f(0)=2`
Put `x=2`. Then,
`f(2)=2+f(1)=4`
Put `x=3`. Then,
`f(3)=3+f(2)=7`
Thus, `f(0),f(1),f(2), …" form a series " 1,2,4,7, …. `
Let `S=1+2+4+7+ … +f(n-1)`
`S=1+2+4+ ... +f(n-2)+f(n-1)`
Subtracting , we get
`0=(1+1+2+3+...+n" terms")-f(n-1)`
` :. f(n-1)=(n(n+1))/(2)`
` :. f(100)=5051`
Promotional Banner

Topper's Solved these Questions

  • RELATIONS AND FUNCTIONS

    CENGAGE ENGLISH|Exercise Solved Examples|15 Videos

  • RELATIONS AND FUNCTIONS

    CENGAGE ENGLISH|Exercise CONCEPT APPLICATION EXERCISE 1.1|15 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE ENGLISH|Exercise Archives (Numerical Value Type)|3 Videos

  • SCALER TRIPLE PRODUCTS

    CENGAGE ENGLISH|Exercise DPP 2.3|11 Videos

Similar Questions

Explore conceptually related problems

Let f: R->R be a continuous function and f(x)=f(2x) is true AAx in R . If f(1)=3, then the value of int_(-1)^1f(f(x))dx is equal to

Let f: R->R be a continuous onto function satisfying f(x)+f(-x)=0AAx in R . If f(-3)=2 \ a n d \ f(5)=4 \ i n \ [-5,5], then the minimum number of roots of the equation f(x)=0 is

Let f be a function satisfying f(x+y)=f(x) + f(y) for all x,y in R . If f (1)= k then f(n), n in N is equal to

Let f(x) = (x)/(1+|x|), x in R , then f is

Let f: R->R be a continuous function and f(x)=f(2x) is true AAx in Rdot If f(1)=3, then the value of int_(-1)^1f(f(x))dx is equal to (a)6 (b) 0 (c) 3f(3) (d) 2f(0)

f:R^+ ->R is a continuous function satisfying f(x/y)=f(x)-f(y) AAx,y in R^+ .If f'(1)=1,then (a)f is unbounded (b) lim_(x->0)f(1/x)=0 (c) lim_(x->0)f(1+x)/x=1 (d) lim_(x->0)x.f(x)=0

Let f_(1) (x) and f_(2) (x) be twice differentiable functions where F(x)= f_(1) (x) + f_(2) (x) and G(x) = f_(1)(x) - f_(2)(x), AA x in R, f_(1) (0) = 2 and f_(2)(0)=1. "If" f'_(1)(x) = f_(2) (x) and f'_(2) (x) = f_(1) (x) , AA x in R then the number of solutions of the equation (F(x))^(2) =(9x^(4))/(G(x)) is...... .

Let f(x)=x^3-6x^2+15 x+3 . Then, (a) f(x)>0 for all x in R (b) f(x)>f(x+1) for all x in R (c) f(x) is invertible (d) f(x)<0 for all x in R

Let f(x) = x - [x] , x in R then f(1/2) is

Let f:(0,oo)->R be a differentiable function such that f'(x)=2-f(x)/x for all x in (0,oo) and f(1)=1 , then f(x) is