Sequence sum formulas (and why they work)

A “sequence sum” means adding the first n terms of a pattern to get one total, written as S_n. In real life, this shows up whenever something grows in steps: saving a little more each month, stacking a repeating discount, or compounding a balance. For arithmetic and geometric sequences, you don’t need to list every term — there are closed‑form formulas that jump straight to the answer.

If each term increases (or decreases) by a fixed amount d, the sequence is arithmetic. With first term a₁, the n-th term is:

• a_n = a₁ + (n − 1)d

The sum of the first n terms is:

• S_n = n/2 · (2a₁ + (n − 1)d)

Intuition: arithmetic sequences have evenly spaced terms. The average of the first and last term is (a₁ + a_n)/2, and there are n terms — so S_n = n · average = n/2 · (a₁ + a_n). Substitute a_n = a₁ + (n−1)d and you get the standard formula.

If each term is multiplied by a fixed ratio r, the sequence is geometric. The n-th term and sum are:

• a_n = a₁ · r^{n − 1}
• If r ≠ 1: S_n = a₁ · (1 − rⁿ) / (1 − r)
• If r = 1: S_n = n · a₁

Intuition: multiply the sum by r, then subtract; almost everything cancels, leaving a simple expression you can divide by (1−r). That “cancellation trick” is why geometric series are so friendly once you know the formula.

When |r| < 1, geometric terms shrink, and the sum approaches a limit: S∞ = a₁/(1−r). That’s the math behind “keep halving a distance forever” puzzles. This page focuses on finite S_n, but it will display that limit when valid so you can compare.

• “Add the same amount each time” → arithmetic (difference d).
• “Multiply by the same percent/factor” → geometric (ratio r = 1 + percent).
• “Total after n steps/payments/terms” → S_n.
• “Value at step n” → a_n.

Real examples you can copy

Try these exactly as written, then adjust the sliders to see how each parameter changes the sum. If you’re studying, a good habit is to compute a few early terms by hand first — it helps you spot input mistakes.

a₁ = 3, d = 2. For n = 10: a₁₀ = 3 + 9·2 = 21 and S₁₀ = 10/2 · (2·3 + 9·2) = 120.

20, 17, 14, 11, … has a₁ = 20, d = −3. For n = 6: a₆ = 5 and S₆ = 75.

3, 6, 12, 24, … has a₁ = 3, r = 2. For n = 8: a₈ = 384 and S₈ = 765.

100, 50, 25, … has a₁ = 100, r = 0.5. For n = 10: S₁₀ ≈ 199.8047, and as n increases it approaches 200.

Suppose a value starts at 100 and grows by 1% each step. That’s geometric with a₁ = 100 and r = 1.01. Compare n = 12 vs n = 120. The n-th term grows slowly at first, then noticeably over long horizons. This is the simplest sandbox for understanding compound growth.

A viral comparison: arithmetic growth feels steady; geometric growth can explode (or vanish) with small ratio changes. Slide r from 0.95 → 1.05 and watch the sum and the meter react.

Sequence sum formulas (and why they work)

A “sequence sum” means adding the first n terms of a pattern to get one total, written as S_n. In real life, this shows up whenever something grows in steps: saving a little more each month, stacking a repeating discount, or compounding a balance. For arithmetic and geometric sequences, you don’t need to list every term — there are closed‑form formulas that jump straight to the answer.

If each term increases (or decreases) by a fixed amount d, the sequence is arithmetic. With first term a₁, the n-th term is:

• a_n = a₁ + (n − 1)d

The sum of the first n terms is:

• S_n = n/2 · (2a₁ + (n − 1)d)

Intuition: arithmetic sequences have evenly spaced terms. The average of the first and last term is (a₁ + a_n)/2, and there are n terms — so S_n = n · average = n/2 · (a₁ + a_n). Substitute a_n = a₁ + (n−1)d and you get the standard formula.

If each term is multiplied by a fixed ratio r, the sequence is geometric. The n-th term and sum are:

• a_n = a₁ · r^{n − 1}
• If r ≠ 1: S_n = a₁ · (1 − rⁿ) / (1 − r)
• If r = 1: S_n = n · a₁

Intuition: multiply the sum by r, then subtract; almost everything cancels, leaving a simple expression you can divide by (1−r). That “cancellation trick” is why geometric series are so friendly once you know the formula.

When |r| < 1, geometric terms shrink, and the sum approaches a limit: S∞ = a₁/(1−r). That’s the math behind “keep halving a distance forever” puzzles. This page focuses on finite S_n, but it will display that limit when valid so you can compare.

• “Add the same amount each time” → arithmetic (difference d).
• “Multiply by the same percent/factor” → geometric (ratio r = 1 + percent).
• “Total after n steps/payments/terms” → S_n.
• “Value at step n” → a_n.

Real examples you can copy

Try these exactly as written, then adjust the sliders to see how each parameter changes the sum. If you’re studying, a good habit is to compute a few early terms by hand first — it helps you spot input mistakes.

a₁ = 3, d = 2. For n = 10: a₁₀ = 3 + 9·2 = 21 and S₁₀ = 10/2 · (2·3 + 9·2) = 120.

20, 17, 14, 11, … has a₁ = 20, d = −3. For n = 6: a₆ = 5 and S₆ = 75.

3, 6, 12, 24, … has a₁ = 3, r = 2. For n = 8: a₈ = 384 and S₈ = 765.

100, 50, 25, … has a₁ = 100, r = 0.5. For n = 10: S₁₀ ≈ 199.8047, and as n increases it approaches 200.

Suppose a value starts at 100 and grows by 1% each step. That’s geometric with a₁ = 100 and r = 1.01. Compare n = 12 vs n = 120. The n-th term grows slowly at first, then noticeably over long horizons. This is the simplest sandbox for understanding compound growth.

A viral comparison: arithmetic growth feels steady; geometric growth can explode (or vanish) with small ratio changes. Slide r from 0.95 → 1.05 and watch the sum and the meter react.