Sequence Calculator - Arithmetic and Geometric nth Term and Sum

Four boxes, then Calculate. This sequence calculator handles both classic pattern types in one form, so start with the dropdown.

• Sequence type - the switch. Pick Arithmetic (common difference) and the form behaves as an arithmetic sequence calculator. Pick Geometric (common ratio) instead and it becomes a geometric sequence calculator.
• First term ($a_1$) - where the pattern starts, such as 3. Every sequence calculator run grows from this box.
• Common difference or ratio - one box, two jobs. In Arithmetic mode it holds $d$, the number the sequence calculator adds each step. In Geometric mode it holds $r$, the multiplier the sequence calculator applies instead.
• Term number (n) - which term you want, and the sequence calculator sums exactly that many terms.

Three rows come back from the sequence calculator. The nth term $a_n$ is the value of term n. The sum of the first n terms $S_n$ adds everything from $a_1$ through $a_n$. The Check row must read true - it confirms the sequence calculator avoided the one forbidden ratio, $r = 1$. The sequence calculator prints it as one word: true.

Watch the pre-set demo values once. Type 3, 2, and 5 in Arithmetic mode, and the sequence calculator returns $a_5 = 11$ and $S_5 = 35$ on the spot. Switch to Geometric with 2, 3, and 4, and the same sequence calculator returns $a_4 = 54$ and $S_4 = 80$ just as fast. Every worked example below traces back to these sequence calculator rows.

This page is more than a static sequence calculator - it is also an AI tutor you can ask questions:

• Ask in plain words. "My sequence starts at 3 and adds 2 each time - what is the 10th term?" The AI fills the form and runs this sequence calculator for you.
• Paste your terms. Type 4, 12, 36, 108 into the question box, and the AI names the pattern and sets the sequence calculator dropdown.
• Ask about the ideas, not just the number. "Why is the exponent n - 1?" The AI explains step by step - every number still comes from the calculator on this page, so nothing is guessed.
• Work backwards. Give a term and the ratio, then ask for the first term instead. The sequence calculator covers both directions.
• Keep asking follow-ups. Ask "why?" as often as you need - the sequence calculator page keeps teaching.

Some patterns add - and that split is exactly why one sequence calculator can serve both types. The sequence 3, 5, 7, 9, 11 adds 2 at every step. The blue bars below grow by that same slice each time. Pick any term, subtract the one before it, and you get the same number every time. That constant is the common difference, written $d$. Arithmetic mode on the sequence calculator is built for exactly this row of numbers.

The idea is old enough to quote. An 1866 algebra text says: "An Arithmetical Progression is a series of quantities which increase or decrease by a common difference." The decrease half matters too. The row 12, 9, 6, 3 steps down by $d = -3$, and the sequence calculator takes a negative difference without any special setup.

Arithmetic sequences are sometimes called arithmetic progressions, or A.P.s - both names mean the add-a-constant pattern, and the sequence calculator's Arithmetic mode answers to either. Now watch the violet bars in the picture outrun the blue ones as n grows. That is the second pattern, and the sequence calculator keeps it one dropdown away.

Other patterns multiply. The violet bars above start at 2 and triple each step: 2, 6, 18, 54. Divide any term by the one before it, and the quotient never changes. That constant is the common ratio, written $r$, and it defines a geometric sequence. Geometric mode on the sequence calculator multiplies this way.

The classic first example is doubling: 1, 2, 4, 8, 16, and so on, with $r = 2$. Ratios do not have to be whole numbers. Halving gives 4, 2, 1, 0.5 with $r = 0.5$ - the sequence calculator accepts fractions there too. A salary that grows 2% a year multiplies by 1.02 each year; $26,000 reaches about$27,591 after three years. That relentless multiplication is geometric growth, and the geometric sequence calculator mode takes any ratio: 3, 0.5, even a negative one. Any of them goes straight into the sequence calculator's ratio box.

One special value: $r = 0$. The sequence collapses to its first term followed by zeros, and the sequence calculator prints exactly that. Geometric sequences are sometimes called geometric progressions, or G.P.s - the sequence calculator answers to both names.

Both patterns, four formulas, one form - every letter matches a field of the sequence calculator form.

Arithmetic sequence formulas

$a_n = a_1 + (n - 1)d$

$S_n = \frac{n}{2}\left(2a_1 + (n - 1)d\right) = \frac{n}{2}(a_1 + a_n)$

The second form is the pairing trick. Write the sum forwards, then write it again backwards; every column adds to the same total. An 1866 rulebook phrases it as "multiply half the sum of the two extremes by the number of terms." For 3, 5, 7, 9, 11 that gives $(3 + 11)/2 \times 5 = 35$. This pair powers the arithmetic sequence calculator mode. Ray's era did it by hand; the sequence calculator does it on click.

Geometric sequence formulas

$a_n = a_1 r^{n-1}$

$S_n = a_1 \frac{1 - r^{n}}{1 - r}, \quad r \neq 1$

The $n - 1$ is not decoration. The first term carries no ratio yet - it is $a_1 \times r^0$ - so the ratio count starts one step behind the term count. When $|r| < 1$ the partial sums approach a ceiling, $a_1/(1 - r)$; halving from 3 creeps toward $3/(1 - 0.5) = 6$ and never passes it. The sequence calculator's S row creeps the same way. The geometric sequence calculator mode guards the $r \neq 1$ rule with its Check row.

Match a letter, fill a box, and the sequence calculator does the rest.

Every number the sequence calculator prints is one of these four lines in disguise. Not sure which pattern you have? Subtract neighboring terms first; a constant difference means arithmetic. If subtraction fails, divide instead - a constant ratio means geometric. The sequence calculator needs that one dropdown choice, and this test makes it for you.

Give the sequence calculator its home numbers in Arithmetic mode: $a_1 = 3$, $d = 2$, $n = 5$.

1. Write the terms: 3, 5, 7, 9, 11 - each step adds the same $d = 2$.
2. Nth term: $a_5 = 3 + (5 - 1)\cdot 2 = 3 + 8 = 11$.
3. Sum: $S_5 = \frac{5}{2}(2 \cdot 3 + 4 \cdot 2) = \frac{5}{2} \cdot 14 = 35$.
4. Pairing check: first plus last is $3 + 11 = 14$, so $S_5 = \frac{5}{2} \cdot 14 = 35$. Same answer twice.

Type 3, 2, and 5 into the sequence calculator above with Arithmetic selected - the sequence calculator confirms 11 and 35 instantly.

Switch the sequence calculator to Geometric mode: $a_1 = 2$, $r = 3$, $n = 4$.

1. Write the terms: 2, 6, 18, 54 - each step multiplies by $r = 3$.
2. Nth term: $a_4 = 2 \cdot 3^{3} = 2 \cdot 27 = 54$.
3. Sum: $S_4 = 2 \cdot \frac{1 - 3^{4}}{1 - 3} = 2 \cdot \frac{-80}{-2} = 80$.
4. Check by hand: 2 + 6 + 18 + 54 = 80, and the sequence calculator agrees.

Now run it backwards. Suppose the fourth term is 54 and the ratio is 3, but the start is missing. Divide out the ratio: 54 is the start times $3^3 = 27$, so the start is $54/27 = 2$. The same dividing move finds $n$ whenever you know a term and the first term - then the sequence calculator takes over again.

Enter 2, 3, and 4 in Geometric mode - the sequence calculator returns 54 and 80 in one click.

Keep the sequence calculator in Geometric mode with a fractional ratio: $a_1 = 3$, $r = 0.5$, $n = 5$.

1. Write the terms: 3, 1.5, 0.75, 0.375, 0.1875 - each step multiplies by 0.5.
2. Nth term: $a_5 = 3 \cdot 0.5^{4} = 0.1875$.
3. Sum: $S_5 = 3 \cdot \frac{1 - 0.5^{5}}{1 - 0.5} = 3 \cdot \frac{0.96875}{0.5} = 5.8125$.

A ratio between 0 and 1 shrinks the terms instead of growing them, and the partial sums creep toward a ceiling. Here the ceiling is $3/(1 - 0.5) = 6$, and 5.8125 is still one small step short of it. The sequence calculator never crosses it either. Type 3, 0.5, and 5 into the sequence calculator above - the sequence calculator's S row lands exactly on 5.8125.

Three slips the sequence calculator catches every day:

1. Adding when the pattern multiplies. Test before you compute: subtract neighboring terms, and if that is not constant, divide them instead. A constant ratio means geometric. Feed a ratio to the arithmetic setup and every answer is quietly wrong - the sequence calculator only knows what the dropdown says.
2. Using n instead of n - 1. The first term already holds the starting value, so the step count is one behind the term count. With $a_1 = 3$ and $d = 2$, writing $3 + 5 \cdot 2 = 13$ jumps the gun; the fifth term is $3 + 4 \cdot 2 = 11$. The sequence calculator uses the same $n - 1$ in both modes, for the same reason.
3. A geometric sum with r = 1. The sum formula divides by $1 - r$, which is zero when $r = 1$. That sequence is constant - every term equals $a_1$ - so the honest total is $n \cdot a_1$. The Check row in the sequence calculator reads false and warns you before you trust a bad number.