Sum of an unknown sequence (perhaps arithmetic or geometric)

Question

The problem is stated as thus:

Given a sequence $(a_n)_{n\geq 1}$,

$$ a_1+a_2+\dots+a_n = 1 + 2^{n+1} $$ for all $n$. Find $a_5$.

This is all the information that is given.

Answer 1

This can be done quite easily. Let us first find $a_{1}.$ By the definition of the sequence, we have $a_{1} = 1 + 2^{2} = 5.$ We can again use the definition to see that $5 + a_{2} = 1 + 2^{3} = 9,$ so $a_{2} = 4.$ Continue as follows: $$5 + 4 + a_{3} = 2^{4} + 1 = 17$$ $$a_{3} = 8$$ $$5 + 4 + 8 + a_{4} = 2^{5} + 1 = 33$$ $$a_{4} = 16$$ $$5 + 4 + 8 + 16 + a_{5} = 2^{6} + 1 = 65,$$ so we have $$a_{5} = \boxed{32}.$$

Answer 2

Using your relation for $n=4$ and $n=5$: $$\begin{align} a_1+a_2+a_3+a_4= 1+2^{4+1} &= 33\\ a_1+a_2+a_3+a_4+a_5= 1+2^{5+1} &= 65 \end{align}$$ So you get $$ a_5 = (a_1+a_2+a_3+a_4+a_5)-(a_1+a_2+a_3+a_4) = 65-33 = 32 $$