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

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

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

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

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

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

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

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

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.

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

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