Use mathematical induction to prove that each statement is true for every positive integer n. ${\sum }_{i=1}^{n}5\cdot 6^i=6(6^n-1)$

Verified step by step guidance

1. Step 1: Understand the statement to prove by induction. We want to prove that for every positive integer \(n\), the sum \(\sum_{i=1}^n 5 \cdot 6^i = 6(6^n - 1)\) holds true.
2. Step 2: Base Case: Verify the statement for \(n=1\). Substitute \(n=1\) into both sides of the equation and check if they are equal.
3. Step 3: Inductive Hypothesis: Assume the statement is true for some positive integer \(k\), that is, assume \(\sum_{i=1}^k 5 \cdot 6^i = 6(6^k - 1)\).
4. Step 4: Inductive Step: Using the inductive hypothesis, prove the statement for \(k+1\). Start with \(\sum_{i=1}^{k+1} 5 \cdot 6^i = \left( \sum_{i=1}^k 5 \cdot 6^i \right) + 5 \cdot 6^{k+1}\) and substitute the inductive hypothesis into this expression.
5. Step 5: Simplify the right-hand side expression after substitution and show that it equals \$6(6^{k+1} - 1)$, thus completing the induction proof.

Key Concepts

Mathematical Induction

Mathematical induction is a proof technique used to verify statements for all positive integers. It involves two steps: proving the base case (usually n=1) is true, and then proving that if the statement holds for an integer k, it also holds for k+1. This method establishes the truth of the statement for all n.

Summation of Geometric Series

A geometric series is a sum of terms where each term is a constant multiple (common ratio) of the previous one. The formula for the sum of the first n terms of a geometric series with first term a and ratio r (r ≠ 1) is a(r^n - 1)/(r - 1). Recognizing this helps simplify and prove the given summation.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using properties of exponents and arithmetic operations. In this problem, it is essential to rewrite sums and expressions to match the form on the right side of the equation, facilitating the induction step and verification.