Arithmetic and geometric sequence calculator
General term, sum of terms and infinite sum of an arithmetic or geometric sequence, with the starting index of the textbook (u₀ or u₁).
Result
35
- Calculation
- un = 5 + 10 × 3 = 35
- Sum from u₀ to u₁₀
- 220
- Number of terms
- 11
- Monotonicity
- increasing
Calculation, assumptions and sources
OverviewThe first term, its index (u₀ or u₁), the common difference or ratio and the target index give the general term, the sum of terms up to it, the monotonicity and, for a geometric sequence with |q| < 1, the infinite sum.
The first term, its index (u₀ or u₁), the common difference or ratio and the target index give the general term, the sum of terms up to it, the monotonicity and, for a geometric sequence with |q| < 1, the infinite sum.
Method & formulasarithmetic: u_n = u_p + (n − p)·r, sum = N·(u_p + u_n)/2
For the arithmetic sum, the first and last terms pair up (Gauss): every pair is worth u_p + u_n, and there are N/2 pairs. The geometric sum comes from S − q·S, which telescopes. For |q| < 1, q^N tends to zero and the infinite sum is u_p/(1 − q), for example 1/2 + 1/4 + 1/8 + … = 1.
- Common difference / ratio
- what separates two consecutive terms: a constant difference (arithmetic, u_{n+1} = u_n + r) or a constant ratio (geometric, u_{n+1} = u_n·q).
- Starting index
- u₀ or u₁ depending on the textbook; it enters the formula through the shift n − p.
- Infinite sum
- the limit of the partial sums. It exists for a geometric sequence with |q| < 1, not for an arithmetic one with non-zero difference.
Validity domainArithmetico-geometric sequences (u_{n+1} = a·u_n + b) and general recurrences are not handled.
Arithmetico-geometric sequences (u_{n+1} = a·u_n + b) and general recurrences are not handled. Indices are zero-or-positive integers.
The index shiftFrom u₀ to u₁₀ there are eleven terms: N = n − p + 1.
From u₀ to u₁₀ there are eleven terms: N = n − p + 1. For a sequence starting at u₁, u₁₀ = u₁ + 9r.