Arithmetic sequence with 12 terms

Every linear pattern boils down to two numbers

An arithmetic sequence adds the same value - the common difference d - to each term. Give the calculator the first term a₁, the difference d, and a position n, and it returns the nth term plus the sum of all terms from a₁ to a_n. The sum uses the Gauss formula: pair the first and last terms, multiply by the count, divide by two.

Quick start

Enter a₁ = 2, d = 5, n = 10. Result: a₁₀ = 47. Sum S₁₀ = 245. Formula: a_n = 2 + (n-1) x 5.

Core formulas

Nth term: a_n = a₁ + (n - 1) x d

Sum of n terms: S_n = n x (a₁ + a_n) / 2

Alternative sum: S_n = n x (2a₁ + (n-1)d) / 2

Common difference: d = a_n - a_n-1 (constant for all pairs)

Reference table - common sequences

Sequence	a₁	d	a₁₀	S₁₀
Natural numbers (1, 2, 3...)	1	1	10	55
Odd numbers (1, 3, 5...)	1	2	19	100
Multiples of 7 (7, 14, 21...)	7	7	70	385
Countdown (100, 90, 80...)	100	-10	10	550
Negative start (-5, -2, 1, 4...)	-5	3	22	85

Practical examples

Saving plan: You save $200 in month 1 and increase by $50 each month. After 12 months: a₁₂ = 200 + 11 x 50 = $750. Total saved: S₁₂ = 12 x (200 + 750) / 2 = $5,700.
An increasing savings plan is an arithmetic sequence

Gauss's trick: Sum of 1 to 100? a₁ = 1, d = 1, n = 100, a₁₀₀ = 100. S₁₀₀ = 100 x (1 + 100) / 2 = 5,050. Legend says young Gauss solved this in seconds while his classmates added one by one.
The pairing trick: 1+100 = 2+99 = 3+98 = ... = 101, fifty pairs

Seats in a theater: Row 1 has 20 seats, each next row adds 2. Row 15: a₁₅ = 20 + 14 x 2 = 48 seats. Total seats in 15 rows: S₁₅ = 15 x (20 + 48) / 2 = 510.
Architecture often uses arithmetic spacing

Depreciation: A machine depreciates by $3,000/year from $45,000. After year 8: a₈ = 45000 + 7 x (-3000) = $24,000. Value hits zero at n = 16 (fully depreciated).
Straight-line depreciation = arithmetic sequence with negative d

Staircase: First step at 18 cm, each next 18 cm higher. Step 14 is at: a₁₄ = 18 + 13 x 18 = 252 cm (2.52 m). Total rise for a standard floor height.
Constant step height = constant common difference

Arithmetic vs geometric - comparison

Property	Arithmetic (this calculator)	Geometric
Pattern	Add d each time	Multiply by q each time
Formula	a_n = a₁ + (n-1)d	a_n = a₁ x q^n-1
Growth	Linear (steady)	Exponential (accelerating)
Graph	Straight line	Exponential curve
Example	Salary +$500/year	Investment x1.08/year

FAQ

What is the common difference?

The common difference d is the constant value added to each term to get the next. In the sequence 5, 11, 17, 23... the difference is d = 11 - 5 = 6. If d is negative, the sequence decreases. If d = 0, every term equals a₁.

How did Gauss sum 1 to 100?

He paired terms from opposite ends: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101... That gives 50 pairs, each summing to 101. Total: 50 x 101 = 5,050. This is exactly the formula S = n(a₁ + a_n)/2. It works for any arithmetic sequence, not just natural numbers.

Can the common difference be a decimal?

Yes. The sequence 1.5, 1.8, 2.1, 2.4... has d = 0.3. All formulas work identically with decimals, fractions, or even irrational differences. The calculator handles any real-number inputs.

How do I find d if I only know two terms?

If you know a_m and a_k (terms at positions m and k): d = (a_m - a_k) / (m - k). Example: a₃ = 11, a₇ = 27. Then d = (27 - 11) / (7 - 3) = 16/4 = 4. Work backward to find a₁: a₁ = a₃ - 2d = 11 - 8 = 3.

When is an arithmetic sequence useful in real life?

Whenever something changes by a fixed amount per period: straight-line depreciation ($5,000/year), fixed salary raises ($500/year), stacking objects (each row adds 2 seats), counting steps (constant riser height), or phone data plans (add 2 GB per tier). If the change is proportional rather than fixed, use a geometric sequence instead.

What happens if n is very large?

The nth term grows linearly: a_n = a₁ + (n-1)d. For d = 1, a_1000000 = 1,000,000. The sum grows quadratically: S_n is roughly n²d/2 for large n. The calculator handles very large n without issues - it uses direct formulas, not iteration.

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Arithmetic sequence of 12 terms - calculate sum and nth term

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