Pythagorist
Algebra

Nth Term Calculator

Find the nth term formula for linear and quadratic sequences. Enter the first few terms of a sequence and the calculator determines whether it is arithmetic (linear) or quadratic, then derives the formula with full step-by-step working.

Enter at least 3 terms for best results

nth term formula

3n +2

How it works

  1. Write the sequence

    Sequence: 5, 8, 11, 14, 17

  2. Find the first differences

    Subtract each term from the next: 3, 3, 3, 3

    Result:1st differences: 3, 3, 3, 3

  3. Constant first differences — linear sequence

    All first differences equal 3, confirming this is a linear (arithmetic) sequence with common difference 3.

  4. Find the nth term formula

    nth term = dn + c, where d = 3 (common difference). To find c, substitute n = 1: 3(1) + c = 5, so c = 2.

    Result:nth term = 3n + 2

  5. Verify

    Check n=1: 3(1)+2 = 5 ✓ n=2: 3(2)+2 = 8 ✓ n=3: 3(3)+2 = 11 ✓

  6. Predict future terms

    Using nth term = 3n +2: n=4 gives 14, n=5 gives 17.

    Result:nth term = 3n +2

The formula

un=an+b(linear)un=an2+bn+c(quadratic)u_n = an + b \quad \text{(linear)} \qquad u_n = an^2 + bn + c \quad \text{(quadratic)}
aaCommon difference (linear) or half the second difference (quadratic)
bbAdjustment constant (found by substituting n = 1)
unuₙThe nth term of the sequence

Example substitution

5,8,11,14,…  ⟹  d=3  ⟹  un=3n+25, 8, 11, 14, \ldots \implies d = 3 \implies u_n = 3n + 2

Common difference is 3. When n=1: 3(1)+c = 5, so c = 2. nth term = 3n + 2.

Worked examples

Find the nth term of 3, 7, 11, 15, …

  1. 1
    1st differences: 4, 4, 4 (constant)
  2. 2
    Common difference d: = 4
  3. 3
    n=1: 4+c=3: c = −1
Answer: nth term = 4n − 1

Find the nth term of 5, 8, 11, 14, …

  1. 1
    1st differences: 3, 3, 3 (constant)
  2. 2
    d = 3:
  3. 3
    n=1: 3+c=5: c = 2
Answer: nth term = 3n + 2

Find the nth term of 1, 4, 9, 16, 25, …

  1. 1
    1st differences: 3, 5, 7, 9 (not constant)
  2. 2
    2nd differences: 2, 2, 2 (constant — quadratic)
  3. 3
    a = 2/2 = 1: nth term = n²
Answer: nth term = n²

Frequently asked questions

How do you find the nth term?+

Find the common difference (d) — this is the coefficient of n. Then find the zero term (value before the first term) — this is the constant. nth term = dn + c.

What is the nth term of 3, 7, 11, 15?+

Common difference = 4. Zero term = 3 − 4 = −1. nth term = 4n − 1.

How do you check an nth term formula?+

Substitute n = 1, 2, 3... and verify the outputs match the original sequence.

What is an arithmetic sequence?+

A sequence where the difference between consecutive terms is constant — this difference is called the common difference.

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