I'm just working on some summer problems so that I can be more prepared when I go into my class in the fall. I found a website full of problems of the content we will be learning but it doesn't have the answers. I need a little guidance on how to do this problem.
In an arithmetic sequence, the third term is $10$ and the fifth term is $16$.
Find the common difference.
Find the first term.
Find the sum of the first $20$ terms in the sequence.
So, the arithmetic formula is $a_n = a_1 + (n – 1)d$ right?
The common difference is the difference between the terms I think. So $16 – 10 = 6$, but there is a term between that so divided by $2$ it is $3$.
How do I find the first term and the sum of the first 20 terms?
Best Answer
The pedestrian approach is to use your formula $a_n=a_1+(n-1)d$ and plug in the two values you know $$a_3=10=a_1+(3-1)d\\a_5=16=a_1+(5-1)d$$ This is two simultaneous equations that you can solve for $a_1,d$. You have already found $d=3$ essentially by subtracting the two. Now just plug that into one of them and evaluate $a_1$.
Otherwise, since you know $d=3$ you can just count two steps down from $a_3=10$ to get $a_1$
To get the sum of the first $20$ terms you need to know (and should have proved) that $$\sum_{i=1}^ni=\frac 12n(n+1)$$ If you write out the sum of the first $20$ terms you will need $d$ times this with $n=19$ plus to account for the $a$s