Question

find the sum of the first 50 terms in the sequence an=3n+2

Answer 1

You know the formula to find out sum of n terms of a series?

Answer 2

\[S_n= \frac{ n }{ 2 }(2a+(n-1)d)\]

Answer 3

You know this formula?

Answer 4

Formula is: first term plus last term, divided by two and times the number of terms:
\[S_n=\frac{ a_1+a_n }{ 2 }·n\]and \[a_n=a_1+(n-1)·d \rightarrow S_n=\frac{ 2·a_1+(n-1)·d }{ 2}·n=a_1·n+\frac{ d }{ 2 }·n·(n-1)\]In your case:
\[a_1=3·1+2=5\] and \[a_{50}=3·50+2=152\]
\[S_{50}=\frac{ 5+152 }{ 2}·50=157·25=3925\]If you apply the second formula, you need to determine d
\[d=a_{n+1}-a_n=3·(n+1)+2-(3·n+2)=3\]
\[S_{50}=50·5+1.5·50·49=250+75·49=3925\]