Please help :))

Question

How does: 4 -((k+2)/(2^k-1)) + (k+1)/(2^k) = 4 - (2(k+2)/(2^k)) + (k+1)/(2^k)

Answer 1

First if all in the 2nd term on the left, it is likely you mean:

(k+2)/(2^(k-1)), that is the "-1" is part of the exponent.

Since the 1st term (4) and the last term (k+1)/2^k on both sides of the equal sign are identical we can focus on how the 2nd terms on both sides of the equal sign are identical.

As a matter of fact the 2nd terms on both sides are prefixed with a minus sign so we can just look at how

(k+2)/2^(k-1) on the left is actually identical to 2(k+2) / 2^k on the right.

Multiply the numerator and denominator of the 2nd term on the left by 2 (which is equivalent to multiplying by 1, which in turn does not change the value of the expression) to see that in fact both terms are identical.

2(k+2)/ 2 (2^(k-1)

The new denominator can be rewritten as 2¹2^(k-1)= 2^(k-1+1)= 2^k.

So by multiplying top and bottom of the 2nd term on the left by 2 we have transformed that term to:

2(k+2) / 2^k which is identical to the 2nd term on the right. This means that the original equation is in fact an identity, true for all values of k.

Here is some corroboration:

desmos.com/calculator/mvdypepq9t