Constructive induction to find a formula for a summation

Question

I am looking to find the values of a b and c for an equation of this summation, but am getting lost on how to solve it. $$\sum_{i=1}^n 12^i = an2^n + b2^n + c$$ $$\sum_{i=1}^{n+1} 12^i = \sum_{i=1}^n 12^i + 12^{n+1}$$ $$\sum_{i=1}^n 12^i + 12^{n+1} = a(n+1)2^{n+1} + b2^{n+1} + c$$ Replace the sum with its equation $$an2^n + b2^n + c + 12^{n+1} = a(n+1)2^{n+1} + b2^{n+1} + c$$ Cancel out the c, multiply the a $$an2^n + b2^n + 12^{n+1} = (an+a)2^{n+1} + b2^{n+1}$$ I know when solving this problem with polynomials you can compare coefficients of like powers to find a and b, but I'm not sure where to go from here.

Answer 1

Using the geometric series $$ \sum_{i=0}^n 12^i = \frac{12^{n+1}-12}{12-1} $$ We have $$ \sum_{i=1}^n 12^i = \frac{12^{n+1}-12}{11} - 1 = \frac{12}{11}\, 12^n-\frac{23}{11} = O(12^n) = O(6^n \cdot 2^n) \overset{!}{=} a \, n \, 2^n + b \, 2^n + c $$ this should grow much faster than any linear combination of $n 2^n$ and $2^n$.

So you can not replace that sum for all values of $n$ with your expression.