If $u_n = a + bn + c2^n$, what's the sum of the first $n$ terms?

If $u_n = a + bn + c2^n$, what's the sum of the first $n$ terms?

Question

The $n$th term of a certain series is of the form $a+bn+c2^n$, where $a$, $b$, and $c$ are numbers. If the first three terms are $2$, $-1$, and $3$, find the values of $a$, $b$, and $c$, and the sum of the first $n$ terms.

Solution

We have

$u_n = a+bn+c2^n,$

so

$$\begin{align} a+b+2c&=2, \\ a+2b+4c&=-1, \\ a+3b+8c&=3. \end{align}$$

Taking $(2)$ $-$ $(1)$ gives $b+2c=-3$, while $(3)$ $-$ $(2)$ gives $b + 4c = 4$.

Subtracting these two equations gives $2c = 7$, and so $c = \dfrac{7}{2}$.

Substituting back into the earlier equations gives $b = -10$, and $a = 5$.

So the $n^{th}$ term $u_n = 5-10n+\dfrac{7}{2}2^n= 5-10n+7\times2^{n-1}.$

Check for the third term: does $3 = 5 - 30 + 28$? Yes.

… and the sum of the first $n$ terms.

This will be

$$\begin{align*} (a + b + 2c) + (a + 2b + 4c) + &(a + 3b + 8c) +...+ (a+nb+2^nc)\\ &= na + b(1 + 2 + .. + n) + c( 2 + 2^2 + 2^3 + ... + 2^n)\\ &= na + b\dfrac{n(n+1)}{2}+ 2c(2^{n}-1)\\ &= 5n -5n^2 - 5n +7\times 2^n -7\\ &= -5n^2 +7\times 2^n -7. \end{align*}$$

Check: when $n= 3$, the sum of the first $n$ terms is $4$. Our formula gives $-45+56-7 = 4$, so we have agreement.