Question Type 5: Find the sum of the first n terms, and find the common difference given the first term and last term Exercises

Given an arithmetic sequence with first term $u_1=3$ and sixth term $u_6=43$, find the common difference $d$.

The 8th term of an arithmetic sequence is $-5$ and the common difference is 3. Calculate the sum of the first 8 terms.

Compute the sum of the first 15 terms of the arithmetic progression where the first term is $10$ and the 15th term is $70$.

For the arithmetic sequence with $u_1=3$ and common difference $d=8$, find the sum of the first 6 terms.

Find $n$ such that the sum of the first $n$ terms of the arithmetic sequence $1,2,3,\dots$ is $210$.

An arithmetic sequence has $u_1=5$ and $u_{10}=50$. Compute the common difference and the sum of the first 10 terms.

The sum of the first 15 terms of an arithmetic sequence is 255 and the common difference is 3. Find the first term.

For the arithmetic series with sum formula $S_n =2n^2 +3n$, find the first term and the common difference.

The sum of the first 30 terms of an arithmetic sequence is $1065$. If the 30th term is $50$, find the first term and the common difference.

In an arithmetic progression the 5th term is 20 and the 12th term is 48. Determine the sum of the first 12 terms.

An arithmetic sequence has $u_1=12$. If the sum of the first $n$ terms equals $390$ and the $n$th term is $48$, find $n$ and $d$.