37.2k views
3 votes
Prove by the principle of Matematical induction3 +3.5+3.5^2+---3.5^2=3(5^n^+^1-1)/4​

1 Answer

6 votes

Final answer:

To prove the given equation using mathematical induction, establish the base case, assume the equation holds for an arbitrary value of n, and prove it for n+1. The equation is proved by adding 3.5^(k+1) to both sides, using the formula for the sum of the geometric series, and simplifying the equation.

Step-by-step explanation:

To prove the given equation using the principle of mathematical induction, we need to first establish the base case and then assume the equation holds for an arbitrary value of n and prove it for n+1.

Base case:

When n = 1, the left side of the equation becomes 3, and the right side becomes 3(5^1+1-1)/4 = 3(5)/4 = 15/4 = 3.75. Thus, the equation holds for n = 1.

Inductive step:

Assume the equation holds for some arbitrary value k, i.e.,

3 + 3.5 + 3.5^2 + ... + 3.5^k = 3(5^k+1-1)/4

We need to prove that it holds for k + 1 as well:

Adding 3.5^(k+1) to both sides of the equation, we get:

3 + 3.5 + 3.5^2 + ... + 3.5^k + 3.5^(k+1) = 3(5^k+1-1)/4 + 3.5^(k+1)

Using the formula for the sum of the geometric series, the left side can be written as:

3(1 - 3.5^(k+1))/(1 - 3.5) + 3.5^(k+1) = 3(1 - 3.5^(k+1))/(1 - 3.5) + 3.5^(k+1)

Simplifying the equation, we get:

3(1 - 3.5^(k+1))/(1 - 3.5) + 3.5^(k+1) = 3(5^(k+1)-1)/4

This proves that the equation holds for k + 1, given that it holds for k.

User Nowaker
by
8.0k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories