Sapewa

## Answered question

2022-01-15

Please, continue the two sequences of numbers below and find an equation to each of the sequences: $n=1,2,3,4,5,6,7$
Equation ${a}_{n}=2,5,9,14,20,27$, ${b}_{n}=1,3,12,60,360,2520$

### Answer & Explanation

Shannon Hodgkinson

Beginner2022-01-17Added 34 answers

You already have an answer for the first, so Ill

RizerMix

Expert2022-01-20Added 573 answers

First sequence: ${a}_{n}=2,5,9,14,20,27,..$ Thus, ${a}_{1}=2$, ${a}_{2}=5$, ${a}_{3}=9$, ${a}_{4}=14$ Now, ${a}_{2}-{a}_{1}=3$ ${a}_{3}-{a}_{2}=4$ ${a}_{4}-{a}_{3}=5$ And so on.. ${a}_{n}-{a}_{n-1}=n+1$ This is cumulative sequence. Thus, we get ${a}_{n}-{a}_{1}=3+4+5+...+\left(n+1\right)$ $⇒{a}_{n}-2=3+4+5+...+\left(n+1\right)$ $⇒{a}_{n}=2+3+4+5+...+\left(n+1\right)$ $⇒{a}_{n}=\frac{n}{2}\left[2+\left(n+1\right)\right]$ Therefore, ${a}_{n}=\frac{n\left(n+3\right)}{2}$ Put $n=7$ ${a}_{7}=\frac{7\left(7+3\right)}{2}=\frac{7×10}{2}=35$ Hence, the equation is ${a}_{n}=\frac{n\left(n+3\right)}{2}$ and ${a}_{7}=35$

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