Using Mathematical Induction to Prove a Summation Formula

AI Thread Summary
The discussion focuses on using mathematical induction to prove the summation formula for \( S_k = 5 \cdot 6 + 5 \cdot 6^2 + \ldots + 5 \cdot 6^k = 6(6^k - 1) \). The next step involves proving \( S_{k+1} \) by adding \( 5 \cdot 6^{k+1} \) to both sides of the equation. The right-hand side is manipulated to show that it can be expressed as \( 6(6^{k+1} - 1) \). Through algebraic manipulation, terms are combined and factored to arrive at the desired form. The proof successfully demonstrates the validity of the summation formula using induction.
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$S_k:5\cdot 6 +5\cdot 6^2+5\cdot 6^3+ ...+5\cdot 6^k=6(6^k-1)$$S_k:5\cdot 6 +5\cdot 6^2+5\cdot 6^3+ ...+5\cdot 6^k+ 5\cdot 6^{k+1}=6(6^k-1)+5\cdot 6^{k+1}$

what do i do now? to prove $S_{k+1}$
 
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The second statement should be labeled as $S_{k+1}$ and you want the right side to be:

$$6\left(6^{k+1}-1\right)$$

Can you get from what you have to there algebraically?

By the way, I am going to move this thread to the Pre-Calculus subforum and retitle it to remove the abbreviation.
 
i don't know how to get there
 
here's all I've done so far:

RHS
$6^{k+1}-6+5\cdot 6^{k+1}$
 
ineedhelpnow said:
here's all I've done so far:

RHS
$6^{k+1}-6+5\cdot 6^{k+1}$

Okay, combine like terms and then factor...
 
$6^{k+1}-6+5\cdot 6^{k+1}$

$[6^{k+1}+5\cdot 6^{k+1}-6$

$[6^{k+1}(1+5)]-6$

$6[6^k(6)-1]$

$6(6^{k+1}-1)$
 
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