Maths · 10-12

Arithmetic Progressions and Series

An arithmetic progression (AP) is a sequence where consecutive terms have a constant difference (common difference d). Arithmetic series is the sum of

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What is the difference between the nth term formula and the sum formula?

The nth term formula a_n = a + (n-1)d finds a specific individual term. The sum formula S_n = n/2 × (2a + (n-1)d) finds the total of all terms from the first to the nth.

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Can an AP have negative common difference?

Yes. If d < 0, the sequence decreases. For example, 100, 95, 90, 85, ... has d = -5 and is a valid AP. It approaches negative infinity as n increases.

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How do you find the common difference if you only know two terms and their positions?

If a_m and a_n are known (where m < n), then d = (a_n - a_m) / (n - m). For example, if a_3 = 10 and a_7 = 22, then d = (22-10)/(7-3) = 12/4 = 3.

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