Arithmetic and geometric progressions

A geometric progression has ratio q = b₂/b₁. Its nth term is bₙ = b₁qⁿ⁻¹. For q ≠ 1, the first n terms sum to Sₙ = b₁(qⁿ − 1)/(q − 1); for q = 1, Sₙ = nb₁.

Calculate differences between several consecutive terms and, when the preceding term is nonzero, their ratios. A constant difference indicates an arithmetic progression; a constant ratio indicates a geometric progression. Check every pair.

First find the term at the indicated index. The remaining sequence is itself geometric with the same ratio, so use that term as its first term in the sum formula.

Express the ratio of the given terms as a power of the common ratio, using the index difference. Take the positive ratio, recover the first term, and apply the sum formula.

Divide the difference of the given terms by the difference of their indices. Recover the first term using the common difference, then apply the sum formula.

An arithmetic progression has difference d = a₂ − a₁. Its nth term is aₙ = a₁ + (n − 1)d. The first n terms sum to Sₙ = n(a₁ + aₙ)/2, or n(2a₁ + (n − 1)d)/2.

A geometric progression has ratio q = b₂/b₁. Its nth term is bₙ = b₁qⁿ⁻¹. For q ≠ 1, the first n terms sum to Sₙ = b₁(qⁿ − 1)/(q − 1); for q = 1, Sₙ = nb₁.

Count the initial drop once. Each later rise has an equally long fall. The rebound heights form a geometric progression; sum them separately from the initial drop.

Write the infinite-series sum formula and rearrange it for the first term. The total includes all later terms, not just the first.

First recover q from the infinite-sum formula. Then apply the general-term formula, taking care with the exponent for the third term.