Answer
please see step-by-step
Work Step by Step
An arithmetic sequence is a sequence whose terms are obtained by adding the same fixed constant $d$ to each term to get the next term.
Thus an arithmetic sequence has the form
$a, a+d, a+2d, a+3d, \ldots$
$a_{n}=a+(n-1)d$
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Let $d_{1}$ be the common difference in the arithmetic sequence $a_{1},\ a_{2},\ a_{3},\ \ldots$,
$a_{n}=a_{1}+(n-1)d_{1}$,
Let $d_{2}$ be the common difference for $b_{1},\ b_{2},\ b_{3},\ \ldots$,
$b_{n}=b_{1}+(n-1)d_{2}$.
Now, observe the sequence $a_{1}+b_{1},\ a_{2}+b_{2},\ \ldots$, where
$a_{n}+b_{n}=[a_{1}+(n-1)d_{1}]+[b_{1}+(n-1)d_{2}]$
$a_{n}+b_{n}=(a_{1}+b_{1})+(n-1)(d_{1}+d_{2})$,
So,
$a_{1}+b_{1},\ a_{2}+b_{2},\ \ldots$
is an arithmetic sequence with first term $a_{1}+b_{1}$
and common difference $d_{1}+d_{2}$.
