What the last term added

The idea

The nth term of an AP is the difference of the sums of its first n terms and first (n – 1) terms: aₙ = Sₙ – Sₙ₋₁.

Running totals

A piggy bank has ₹21 after 3 days and ₹10 after 2 days.

You want to know how many rupees were put in on day 3.

How can you find it from the two totals?

• Subtract: 21 − 10
• Add: 21 + 10
• Divide 21 by 3

Find the terms from the totals

Only the running totals S₁, S₂, S₃, … are shown. Tap a ? to find that term. The bars show the two totals you subtract.

Last term = difference of sums

The nth term of an AP is the sum of the first n terms minus the sum of the first (n − 1) terms: aₙ = Sₙ − Sₙ₋₁.

Sₙ is just Sₙ₋₁ with one more term, aₙ, put on the end. So taking Sₙ₋₁ away from Sₙ leaves aₙ. For the first term, S₀ = 0, so a₁ = S₁.

Notes

aₙ = Sₙ − Sₙ₋₁. The sum of n terms is the sum of n − 1 terms with aₙ added on. For the first term, a₁ = S₁.

Check yourself

The totals of an AP are S₇ = 105 and S₈ = 136. Find the 8th term.

Answer: 31

a₈ = S₈ − S₇ = 136 − 105 = 31.

For an AP, the sum of the first n terms is Sₙ = n² + 2n. Find the 3rd term.

Answer: 7

S₃ = 3² + 2 × 3 = 15 and S₂ = 2² + 2 × 2 = 8. So a₃ = 15 − 8 = 7.

Which is true for the nth term of an AP (for n ≥ 2)?

• aₙ = Sₙ − Sₙ₋₁ — correct. Yes! Sₙ has one more term than Sₙ₋₁, and that extra term is aₙ.
• aₙ = Sₙ + Sₙ₋₁. Adding the two sums counts the first n − 1 terms twice. Subtract instead.
• aₙ = Sₙ ÷ n. Sₙ ÷ n is the average of the first n terms, not the nth term.