How to turn this (2^(k+3))+(k⋅(2^(k+2)))+(k⋅(2^(k+2))+2) into (k+1)(2^(k+3)+2)?
I know I need to factor out (2^(k+2)) of the middle two terms, then simplify, and then factor (2^(k+3)) out of the term you get. But I'm still at lost. PLEASE someone, I NEED your help!
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2^(k+3) = 2^(k+2) * 2^1
2^(k+2) *2 + 2^(k+2) k +2^(k+2) k +2
2 * 2^(k+2) +2k * 2^(k+2) +2
2^(k+2) [2k+2] +2
2(k+1) *2^(k+2) +2 =
[(k+1) * 2^(k+3)] +2
(2^(k+3))+(kâ (2^(k+2)))+(kâ (2^(k+2))+2)
That sure is a lot of parentheses. Addition is associative, so let's get rid of a few:
2^(k+3)+kâ 2^(k+2)+kâ 2^(k+2)+2
Notice we have two instances of the term "kâ 2^(k+2)", so:
= 2^(k+3)+2kâ 2^(k+2)+2
But 2â 2^n = 2^(n+1), so:
= 2^(k+3)+kâ 2^(k+3)+2
Use the distributive property to factor out 2^(k+3):
= (k+1)(2^(k+3))+2
And you're there.