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.