Arithmetic Sequence

00:00 5 marks

An arithmetic sequence has fifth term \(u_5 = 11\) and twelfth term \(u_{12} = 25\).
(a) Find the first term \(u_1\) and the common difference \(d\). [3]
(b) Find the sum of the first 20 terms of the sequence, \(S_{20}\). [2]

Hint Hint

Hint

Set up the equations \(u_1 + 4d = 11\) and \(u_1 + 11d = 25\), subtract them to solve for \(d\), then use the sum formula \(S_n = \frac{n}{2}(2u_1 + (n-1)d)\) to calculate \(S_{20}\).

Answer Answer
Mark Scheme Mark Scheme
Explanation Explanation
Examiner Tips Examiner Tips
Common Errors Common Errors
Real World Context Real World Context

Answer Answer

Complete Answer

a) From \(u_5 = u_1 + 4d = 11\) and \(u_{12} = u_1 + 11d = 25\), subtracting gives \(7d = 14\) so \(d = 2\) and substituting back gives \(u_1 = 3\).

b) Using the sum formula \(S_{20} = \frac{20}{2}(2(3) + 19(2)) = 10(6 + 38) = 10\times44 = 440\).

Mark Scheme Mark Scheme

Answer Explanation Answer Explanation

Examiner Tips Examiner Tips

Common Errors Common Errors