Question:

If $f(1)=1$ and $f(n+1)=2f (n)+1$ if $n \ge1$, then $f(n)$ is equal to

Updated On: Jul 6, 2022
  • $2^{n}+1$
  • $2^{n}$
  • $2^{n}-1$
  • $2^{n-1}-1$
Hide Solution
collegedunia
Verified By Collegedunia

The Correct Option is C

Solution and Explanation

$f\left(1\right)=1$ and $f\left(n+1\right)=2f\left(n\right)+1, n \ge1$. $\therefore f\left(2\right)=2\left(1\right)+1=3, f\left(3\right)=7, f\left(4\right)=15, \ldots$ and so on Thus, $f\left(1\right)=2^{1}-1, f\left(2\right)=2^{2}-1$, $f\left(3\right) = 2^{3} - 1, \ldots, f\left(n\right)$ $=2^{n}-1$.
Was this answer helpful?
0
0

Top Questions on Relations and functions

View More Questions

Notes on Relations and functions

Concepts Used:

Relations and functions

A relation R from a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.

A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. In other words, no two distinct elements of B have the same pre-image.

Representation of Relation and Function

Relations and functions can be represented in different forms such as arrow representation, algebraic form, set-builder form, graphically, roster form, and tabular form. Define a function f: A = {1, 2, 3} → B = {1, 4, 9} such that f(1) = 1, f(2) = 4, f(3) = 9. Now, represent this function in different forms.

  1. Set-builder form - {(x, y): f(x) = y2, x ∈ A, y ∈ B}
  2. Roster form - {(1, 1), (2, 4), (3, 9)}
  3. Arrow Representation