Geometric Series Applications

This lesson covers advanced applications of geometric series, including proof of the sum formula, convergence analysis, and real-world modelling. These skills are frequently examined in Edexcel A-Level Mathematics (9MA0).

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Proof of the Sum Formula

Prove that S(n) = a(1 - r^n) / (1 - r) for r ≠ 1.

Write the sum: S(n) = a + ar + ar² + ar³ + ... + ar^(n-1)

Multiply both sides by r: r x S(n) = ar + ar² + ar³ + ... + ar^(n-1) + ar^n

Subtract the second from the first: S(n) - r x S(n) = a - ar^n

S(n)(1 - r) = a(1 - r^n)

Since r ≠ 1: S(n) = a(1 - r^n) / (1 - r)

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Proof of the Sum to Infinity Formula

Prove that for |r| < 1, S(infinity) = a / (1 - r).

Starting from S(n) = a(1 - r^n) / (1 - r):

When |r| < 1, as n tends to infinity, r^n tends to 0.

Therefore: S(n) tends to a(1 - 0) / (1 - r) = a / (1 - r)