Sequences and Series

Let $a_1, a_2, a_3, \ldots$ be a geometric progression of positive integers such that $a_1 = 3$ and $a_{n+2} - 2a_n = a_{n+1}$ for all positive integers $n$. What is the value of $a_1 + a_2 + a_3 + a_4 + a_5$?

Let $S_n$ denote the sum of the first $n$ terms of a sequence $a_1, a_2, a_3, \ldots$. If $S_{n+3}-S_n=13 n+7$ for all $n$, what is the value of $a_{13}-a_{10}$ ?

Let $a_1, a_2, a_3, \ldots$ be a sequence of real numbers. Let $s_n=a_1+a_2+\cdots+a_n$. If $2 s_n=$ $n\left(c+a_n\right)$ for some real number. for some real number $c$ and for all $n=1,2,3, \ldots$, then which one of the following stateme...

A lab reports that the global average temperature in the year 2020 was $14.9^{\circ} \mathrm{C}$ and predicts that the global average temperature will increase at the rate of $1 \%$ per year. What will be the global average temperature in t...