 Lewis Harvey

2021-02-19

Write a recursive rule and an explicit rule for an arithmetic sequence that models a situation. Then use the rule to answer the question. Jack begins an exercise routine for 10 minutes each day. Each week he plans to add 10 minutes per day to his exercise routine. For how many minutes will he exercise for each day on the 7th week? Maciej Morrow

If Jack exercises for 10 minutes for the first day, then the first term of the sequence is ${a}_{1}=10$.
If he adds 10 minutes to his exercise routine each day, then the sequence is arithmetic since it has a common difference of d=10.
The recursive formula for an arithmetic sequence is ${a}_{n}={a}_{n-1}+d$, $n\ge 2$.

Since $d=10$, the the recursive formula is then ${a}_{1}=10$, ${a}_{n}={a}_{n-1}+10,n\ge 2$
The explicit formula for an arithmetic sequence is ${a}_{n}={a}_{1}+\left(n-1\right)d$.

Since ${a}_{1}=10$ and d=10, then ${a}_{n}=10+\left(n-1\right)\left(10\right)$.
Disturbing the 10 gives ${a}_{n}=10+{10}_{n}-10$. Combining like terms then gives ${a}_{n}={10}_{n}$.
The 7th week is when $n=7$. Substituting $n=7$ into ${a}_{n}={10}_{n}$ the gives ${a}_{7}=10×7=70$. He will then exercise for 70 min in the 7th week.

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