Question

find the sum. 7+10+13+...+97

Answer 1

The sequence 7,10,13,...97 is a arithmetic sequence, with
\[\Large a_1=7\\
\Large d=3\]
And if n is the number of the term 97 then :
\[\Large 97= a_n=a_1+(n-1)d=7+3(n-1)\]
So :
\[\Large 3(n-1)=97-7=90\]
So :
\[\Large n-1=30\]
So :
\[\Large n=31\]
Now to find the sum we have :
\[\Large 7+10+13+\cdots+97=a_1+a_2+\cdots+a_{31}
\\\Large~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{31}{2}(a_1+a_{31})
\\\Large~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{31}{2}(3+97)
\\\Large~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=1550
\]