If (5^(k-1))+(5^(k+1)) = m, what is 2*5^k in terms of m?

Question

If 5^k-1+5^k+1=m, what is 2·5^k in terms of m?

A)5m

B)5m÷2

C) 5m÷13

D) 5m÷26

Answer 1

If 5^k-1+5^k+1=m, what is 2·5^k in terms of m?

First let’s solve for 5^k

5^k-1 + 5^k+1 = m

For 5^k-1 the exponent is subtracted so this is the same as 5^k/5¹ or 5^k/5 or 5^k·1/5

For 5^k+1 the exponent is added so this is the same as 5^k·5¹ or 5^k·5

Let’s substitute these back into our equation:

5^k-1 + 5^k+1 = m is the same as

(5^k·1/5) + (5^k·5) = m

Let’s factor out 5^k from both sets of parentheses

5^k (1/5 + 5) = m now simplify what we have left in the parentheses

5^k (1/5 + 25/5) = m

5^k (26/5) = m

Multiply both sides by 5/26 to cancel out the fraction and we have

5^k = m · 5/26 or

5^k = 5m/26

So…. Finally, to get 2·5^k we multiply both sides of the equation by 2

2·5^k = 2(5m/26)

2·5^k = 10m/26 then reduce the fraction

2·5^k = 5m/13

Answer 2

5^{k -1} + 5^k+1 = m; 5^k(1/5 + 5) = m; 2·5^k·(26/5) = 2m; 2·5^k = 5m/13. Answer is C)