Arithmetic and geometric sequences
arithmetic: u_n = u_p + (n − p)·r, sum = N·(u_p + u_n)/2
Enter the first term, its index (u₀ or u₁, following your textbook), the common difference or ratio and the target index: the tool computes the general term, the sum of terms up to it, the monotonicity, and, for a geometric sequence with |q| < 1, the exact infinite sum, with the honest verdict “does not exist” when |q| ≥ 1.
35
- Calculation
- u_n = 5 + 10 × 3 = 35
Scientific dossier
What the tool computes, what it assumes, where it stops being valid, and where its data comes from.
Method & formulasarithmetic: u_n = u_p + (n − p)·r, sum = N·(u_p + u_n)/2
arithmetic: u_n = u_p + (n − p)·r, sum = N·(u_p + u_n)/2
geometric: u_n = u_p·q^(n−p), sum = u_p·(1 − q^N)/(1 − q). Q ≠ 1
infinite sum (|q| < 1): S∞ = u_p / (1 − q)
N = n − p + 1 (number of terms, both bounds included)
The arithmetic sum is Gauss’s trick: pair the first and last terms, every pair is worth u_p + u_n. And there are N/2 pairs. The geometric sum comes from S − q·S, where almost everything telescopes. When |q| < 1, q^N tends to zero and the sum of all the terms converges to u_p/(1 − q), the half-of-a-half paradox: 1/2 + 1/4 + 1/8 + … = 1, exactly.
- 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
- · textbooks start at u₀ or u₁ depending on country and chapter. The formula depends on that choice through the shift n − p: the tool asks instead of imposing, the index error is mistake number one in these exercises.
- Infinite sum
- · the limit of the partial sums as ever more terms are added. It only exists for a geometric sequence with |q| < 1, never for an arithmetic one with non-zero difference.
Validity domainThe tool handles exact arithmetic and geometric sequences, not arithmetico-geometric ones (u_{n+1} = a·u_n + b), nor general recurrences.
The tool handles exact arithmetic and geometric sequences, not arithmetico-geometric ones (u_{n+1} = a·u_n + b), nor general recurrences. For very large indices of a geometric sequence, q^n overflows floating-point capacity around 10³⁰⁸: the result becomes Infinity, which is mathematically the right message. Indices are zero-or-positive integers, as in textbook statements.
Common pitfall: the index shiftFrom u₀ to u₁₀ there are eleven terms, not ten.
From u₀ to u₁₀ there are eleven terms, not ten. Both bounds count (N = n − p + 1). Same trap in the general term: u₁₀ of a sequence starting at u₁ is u₁ + 9r, not u₁ + 10r. It is the fence-post error: ten intervals separate eleven posts. The tool displays the number of terms precisely to catch that shift at a glance.