Question Type 3: Working with arithmetic sequences algebraically Exercises

Find the 10th term of the arithmetic sequence with first term $a_1=5$ and common difference $d=3$.

Given the arithmetic sequence defined by $t_n = 7n - 2$, find its first term and common difference.

An arithmetic sequence has 4th term 11 and 9th term 26. Find its first term and common difference.

The sum of the first 20 terms of an arithmetic sequence is 610 and its first term is 3. Find the common difference.

Given an arithmetic sequence such that $a + d = 10$ and $a + 9d = 46,$ find $a$, $d$, and the sum of the first 15 terms.

In an arithmetic sequence, $t_{n+2} - t_n = 10$ for all $n$, and $t_2=7$. Find the first term and common difference.

In an arithmetic sequence, the 5th term is twice the 3rd term and the 2nd term is 7. Find the first term and common difference.

The sum of the first $n$ terms of an arithmetic sequence is given by $S_n = 3n^2 + 5n.$ Find its first term and common difference.

Two arithmetic sequences share the same first term. The common differences are $d$ and $2d$ respectively. If the 5th term of the first sequence equals the 3rd term of the second sequence plus 4, find $d$.

If three numbers $a$, $ab$, and $a+b$ are consecutive terms of an arithmetic sequence, express $b$ in terms of $a$.

Two arithmetic sequences $(u_n)$ and $(v_n)$ satisfy for all $n$: $u_n + v_n = 5n + 3, \quad u_n - v_n = 2n - 1.$ Find the first terms and common differences of both sequences.

Consider two arithmetic sequences $u_n=an+b$ and $v_n=cn+d$. They satisfy $u_5=v_3$ and $u_{10}=v_6+15$. Given $a=2$ and $c=3$, find $b$ and $d$.