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Arithmetic sequences and series

Find the nnth term using Tn=a+(n−1)dT_n = a + (n-1)d and partial sums using Sn=n2(2a+(n−1)d)S_n = \frac{n}{2}(2a + (n-1)d); identify and work with arithmetic sequences in applied contexts.

Worked examples

Finding the nnth term

Straightforward

Problem

An arithmetic sequence has first term a=3a = 3 and common difference d=5d = 5. Find T10T_{10} and find which term equals 98.

Finding the sum of a series

Moderate

Problem

Find the sum of the arithmetic series 6+11+16+…+966 + 11 + 16 + \ldots + 96.

Using SnS_n to find a term

Challenging

Problem

The sum of the first nn terms of a series is Sn=2n2+nS_n = 2n^2 + n. Show that the series is arithmetic and find the common difference.

Practise

Q1·Straightforward
An arithmetic sequence has first term a=4a = 4 and common difference d=7d = 7. Find the 6th term T6T_6.
Q2·Straightforward
The sequence 3,7,11,15,…3, 7, 11, 15, \ldots is arithmetic. What is the common difference dd?
Q3·Straightforward
The arithmetic sequence 10,3,−4,−11,…10, 3, -4, -11, \ldots has first term a=10a = 10 and common difference d=−7d = -7. Find T4T_4.
Q4·Straightforward
Find the sum of the first 5 terms of the arithmetic series 2+5+8+11+142 + 5 + 8 + 11 + 14.
Q5·Moderate
An arithmetic sequence has first term a=7a = 7 and common difference d=4d = 4. Which term equals 63? Find the value of nn.
Q6·Moderate
The 3rd term of an arithmetic sequence is 11 and the 8th term is 31. Find the common difference dd.
Q7·Moderate
The 3rd term of an arithmetic sequence is 11 and the common difference is 4. Find the first term aa.
Q8·Moderate
An arithmetic series has first term a=5a = 5 and common difference d=3d = 3. Find S10S_{10}.
Q9·Moderate
Find the sum of the arithmetic series 3+8+13+…+1233 + 8 + 13 + \ldots + 123.
Q10·Moderate
A worker earns $1000 in their first week and receives a $50 pay rise each week. What are their total earnings over 8 weeks?
Q11·Challenging
How many terms of the series 1+4+7+…1 + 4 + 7 + \ldots must be added for the sum to first exceed 100?
Q12·Challenging
Three consecutive terms of an arithmetic sequence have a sum of 39 and a sum of squares of 509. Find the largest of the three terms.
Q13·Challenging
The sum of the first nn terms of a series is given by Sn=3n2+2nS_n = 3n^2 + 2n. Find the 7th term T7T_7.