ICSE Class 10 Maths: AP & GP — Sequences & Series Notes 2026

Arithmetic Progression (AP)

• AP: sequence with constant common difference d between consecutive terms
• General term (nth term): Tₙ = a + (n−1)d where a = first term
• Finding n given Tₙ: rearrange formula; identifying AP by checking equal differences

Sum of n Terms of AP

• Sₙ = (n/2)[2a + (n−1)d] or Sₙ = (n/2)(first term + last term)
• Relation between Tₙ and Sₙ: Tₙ = Sₙ − S_{n−1}
• Arithmetic mean of AP = (first term + last term)/2

Geometric Progression (GP)

• GP: sequence with constant common ratio r between consecutive terms
• General term: Tₙ = arⁿ⁻¹ where a = first term
• Identifying GP: ratio Tₙ₊₁/Tₙ = r must be constant

Sum of GP

• Finite GP sum: Sₙ = a(rⁿ−1)/(r−1) for r≠1; Sₙ = na for r=1
• Sum of infinite GP (|r|<1): S∞ = a/(1−r)
• Geometric mean between a and b: G = √(ab)

Inserting Means

• Inserting n arithmetic means between a and b: d = (b−a)/(n+1)
• Inserting n geometric means: r = (b/a)^{1/(n+1)}
• ICSE: insert 3 AMs between 3 and 15; or 3 GMs between 2 and 162

Word Problems — AP and GP

• An AP: first term 3, last term 93, sum = 2010 — find number of terms
• A GP: sum of first 3 terms = 14; sum of next 3 terms = 112 — find first term and ratio
• Population growing at 10% per year: GP model to find population after n years

ICSE Numericals

• Find the sum of first 20 terms of AP: 5, 8, 11, …
• Find sum to infinity of GP: 1 + ½ + ¼ + …
• How many terms of AP 6, 12, 18, … are needed for sum to exceed 300?