Recover the first term and common difference from term information.
Notes and key methods
An arithmetic progression has a constant difference d. Its n-th term is \(u_n=a+(n-1)d\), and \(\displaystyle S_{n}= \frac{n(2a+ (n- 1)d)}{2}\).
Subtract equations for two known terms to find d, then recover a. An index n counts terms and must be a positive integer. Check a recovered index in the original term or sum formula.
Check the conditions
The difference is constant.
A term number is a positive integer.
Three worked examples
Example 1 · Arithmetic progressions
An arithmetic progression has \(\displaystyle u_{2}=9\) and \(\displaystyle u_{5}=21\). Let \(\displaystyle n=9\).
Find a and d, showing your equations.
Find the common difference d.
Find \(\displaystyle u_{n}\).
Find \(\displaystyle S_{n}\).
Show the reasoning
(a)
\(\displaystyle a+d=9\) and \(\displaystyle a+4d=21\); subtract to obtain \(\displaystyle 3d=12\), then recover a.
Answer: \(5\) and \(4\)
(b)
There are three common differences from \(\displaystyle u_{2}\) to \(\displaystyle u_{5}\).
\(4\)
Answer: \(4\).
(c)
Use a+(n−1)d.
\(37\)
Answer: \(37\).
(d)
Use n times the average of the first and last terms.
\(189\)
Answer: \(189\).
Example 2 · Arithmetic progressions
An arithmetic progression has \(\displaystyle u_{2}=11\) and \(\displaystyle u_{5}=26\). Let \(\displaystyle n=6\).
Find a and d, showing your equations.
Find the common difference d.
Find \(\displaystyle u_{n}\).
Find \(\displaystyle S_{n}\).
Show the reasoning
(a)
\(\displaystyle a+d=11\) and \(\displaystyle a+4d=26\); subtract to obtain \(\displaystyle 3d=15\), then recover a.
Answer: \(6\) and \(5\)
(b)
There are three common differences from \(\displaystyle u_{2}\) to \(\displaystyle u_{5}\).
\(5\)
Answer: \(5\).
(c)
Use a+(n−1)d.
\(31\)
Answer: \(31\).
(d)
Use n times the average of the first and last terms.
\(111\)
Answer: \(111\).
Example 3 · Arithmetic progressions
An arithmetic progression has \(\displaystyle u_{2}=8\) and \(\displaystyle u_{5}=11\). Let \(\displaystyle n=7\).
Find a and d, showing your equations.
Find the common difference d.
Find \(\displaystyle u_{n}\).
Find \(\displaystyle S_{n}\).
Show the reasoning
(a)
\(\displaystyle a+d=8\) and \(\displaystyle a+4d=11\); subtract to obtain \(\displaystyle 3d=3\), then recover a.
Answer: \(7\) and \(1\)
(b)
There are three common differences from \(\displaystyle u_{2}\) to \(\displaystyle u_{5}\).
\(1\)
Answer: \(1\).
(c)
Use a+(n−1)d.
\(13\)
Answer: \(13\).
(d)
Use n times the average of the first and last terms.
\(70\)
Answer: \(70\).
Common mistakes
The first term has index 1, so the multiplier is n−1.
Do not confuse the n-th term with the sum of n terms.