Arithmetic Series Applications

This lesson focuses on applying arithmetic series to problem-solving and modelling contexts at A-Level (9MA0). You will also learn to prove the sum formula, a skill Edexcel frequently tests in exam questions.

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Proof of the Sum Formula

This is a classic proof that Edexcel can ask you to reproduce.

Prove that S(n) = n/2 x (2a + (n-1)d)

Write the sum forwards: S(n) = a + (a + d) + (a + 2d) + ... + (a + (n-1)d)

Write the sum backwards: S(n) = (a + (n-1)d) + (a + (n-2)d) + ... + (a + d) + a

Add these two expressions term by term. Each pair sums to: a + (a + (n-1)d) = 2a + (n-1)d

There are n such pairs, so: 2 x S(n) = n x (2a + (n-1)d)

Therefore: S(n) = n/2 x (2a + (n-1)d) as required.

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Using S(n) to Find Individual Terms

A key relationship:

u(n) = S(n) - S(n-1) for n >= 2

This allows you to find any term if you know S(n) as a formula.

Worked Example 1