Arithmetic and geometric series

Objective. Solve multi-condition progression problems.

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Learn: the key idea

Arithmetic and geometric series combine the two standard progression types, arithmetic with constant common difference d and geometric with constant common ratio r, often within the same multi-condition problem requiring simultaneous equations. Example: given an arithmetic sequence's 3rd term is 11 and 7th term is 23, solving a+2d=11 and a+6d=23 gives d=3, a=5. Mixed problems may also ask to compare sums, using Sₙ=n/2(2a+(n−1)d) for arithmetic and Sₙ=a(1−rⁿ)/(1−r) for geometric series.

Worked example

An arithmetic series has first term 4 and common difference 5. A geometric series has first term 4 and common ratio 2. Find the sum of the first 5 terms of each series.

1 $Arithmetic sum: S₅=5/2(2(4)+(5-1)(5))=5/2(8+20)$
2 $Evaluate arithmetic sum: 5/2(28)=70$
3 $Geometric sum: S₅=4(1-2⁵)/(1-2)=4(1-32)/(-1)$
4 $Evaluate geometric sum: 4(-31)/(-1)=124$

Answer: Arithmetic sum 70, geometric sum 124

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Common mistake. Students confuse the two sum formulae, using the geometric sum formula on an arithmetic sequence or vice versa; always check whether terms share a common difference or ratio before selecting a formula.

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