Question

by using Principle of mathematical induction prove that 1+2+4+...+2^(n-1) =2^n-1​

Answer 1

Answer:

Mathematical Induction - Problems With Solutions

Several problems with detailed solutions on mathematical induction are presented.

The principle of mathematical induction is used to prove that a given proposition (formula, equality, inequality…) is true for all positive integer numbers greater than or equal to some integer N.

Let us denote the proposition in question by P (n), where n is a positive integer. The proof involves two steps:

Step 1: We first establish that the proposition P (n) is true for the lowest possible value of the positive integer n.

Step 2: We assume that P (k) is true and establish tha t P (k+1) is also true

Problem 1

Use mathematical induction to prove that

1 + 2 + 3 + ... + n = n (n + 1) / 2

for all positive integers n.

Solution to Problem 1:

Let the statement P (n) be

1 + 2 + 3 + ... + n = n (n + 1) / 2

Step-by-step explanation:

STEP 1: We first show that p (1) is true.

Left Side = 1

Right Side = 1 (1 + 1) / 2 = 1

Both sides of the statement are equal hence p (1) is true.

STEP 2: We now assume that p (k) is true

1 + 2 + 3 + ... + k = k (k + 1) / 2

and show that p (k + 1) is true by adding k + 1 to both sides of the above statement

1 + 2 + 3 + ... + k + (k + 1) = k (k + 1) / 2 + (k + 1)

= (k + 1)(k / 2 + 1)

= (k + 1)(k + 2) / 2

The last statement may be written as

1 + 2 + 3 + ... + k + (k + 1) = (k + 1)(k + 2) / 2

Which is the statement p(k + 1)nation: