Arithmetic and Geometric Sequences

Formulas for the nth term, the sum of the first n terms and the characteristic properties of arithmetic and geometric sequences, plus the sum of an infinite geometric series. Enter your own values in the calculators to check a solution.

11 formulas · Grades 9–10

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All 11 formulas on this page
  1. nth term of an arithmetic sequence
  2. Sum of the first n terms of an arithmetic sequence
  3. nth term of a geometric sequence
  4. Sum of the first n terms of a geometric sequence
  5. Sum of n terms of an arithmetic sequence (via first term and difference)
  6. Characteristic property of a geometric sequence
  7. Characteristic property of an arithmetic sequence
  8. Common difference of an arithmetic sequence (via two arbitrary terms)
  9. nth term of an arithmetic sequence (via an arbitrary kth term)
  10. nth term of a geometric sequence (via an arbitrary kth term)
  11. Sum of an infinite geometric series

nth term of an arithmetic sequence

Grade 9
an=a1+(n−1)da_n = a_1 + (n - 1)d

Lets you find any term of an arithmetic sequence.

Legend:
  • a1 — First term (a1)
  • n — Element number (n)
  • d — Common difference (d)

Sum of the first n terms of an arithmetic sequence

Grade 9
Sn=a1+an2⋅nS_n = \frac{a_1 + a_n}{2} \cdot n

Calculates the sum from the 1st to the nth term of an arithmetic sequence.

Legend:
  • a1 — First term (a1)
  • an — nth term (an)
  • n — Number of elements (n)

nth term of a geometric sequence

Grade 9
bn=b1⋅qn−1b_n = b_1 \cdot q^{n-1}

Lets you find any term of a geometric sequence.

Legend:
  • b1 — First term (b1)
  • q — Common ratio (q)
  • n — Element number (n)

Sum of the first n terms of a geometric sequence

Grade 9
Sn=b1(qn−1)q−1(q≠1)S_n = \frac{b_1 (q^n - 1)}{q - 1} \quad (q \neq 1)

Calculates the sum of the first n terms of a geometric sequence when q ≠ 1.

Legend:
  • b1 — First term (b1)
  • q — Common ratio q (q≠1)
  • n — Count (n)

Sum of n terms of an arithmetic sequence (via first term and difference)

Grade 9
Sn=2a1+(n−1)d2⋅nS_n = \frac{2a_1 + (n-1)d}{2} \cdot n

Calculates the sum of the first n terms of an arithmetic sequence from the first term and the common difference.

Legend:
  • a1 — First term (a1)
  • n — Count (n)
  • d — Common difference (d)

Characteristic property of a geometric sequence

Grade 9
bn2=bn+1⋅bn−1b_n^2 = b_{n+1} \cdot b_{n-1}

The square of a term of a geometric sequence equals the product of its neighbouring terms.

Legend:
  • bnext — Next term (b_{n+1})
  • bprev — Previous term (b_{n-1})

Characteristic property of an arithmetic sequence

Grade 9
an=an−1+an+12a_n = \frac{a_{n-1}+a_{n+1}}{2}

Each term of an arithmetic sequence equals the arithmetic mean of its neighbours.

Legend:
  • aprev — Previous term (a_{n-1})
  • anext — Next term (a_{n+1})

Common difference of an arithmetic sequence (via two arbitrary terms)

Grade 9
d=an+1−an=an−akn−kd = a_{n+1} - a_n = \frac{a_n - a_k}{n - k}

The common difference of an arithmetic sequence from any two of its terms and their indices.

a_ka_na_{n+1}d
d is the step between neighbouring terms, the same throughout the sequence.
Legend:
  • an — Term a_n
  • ak — Term a_k
  • n — Index n
  • k — Index k

nth term of an arithmetic sequence (via an arbitrary kth term)

Grade 9
an=ak+(n−k)da_n = a_k + (n - k)d

Any term of an arithmetic sequence can be found not only from the first term, but from any other known term.

a_ka_n(n − k) шагов по d
From the kth to the nth term there are (n − k) equal steps of d.
Legend:
  • ak — Term a_k
  • n — Index n
  • k — Index k
  • d — Common difference (d)

nth term of a geometric sequence (via an arbitrary kth term)

Grade 9
bn=bk⋅qn−kb_n = b_k \cdot q^{n-k}

Any term of a geometric sequence can be found not only from the first term, but from any other known term.

b_kb_nумножение на q, (n − k) раз
From the kth to the nth term we multiply by q exactly (n − k) times.
Legend:
  • bk — Term b_k
  • n — Index n
  • k — Index k
  • q — Common ratio q

Sum of an infinite geometric series

Grade 10
S=b11−q, ∣q∣<1S = \frac{b_1}{1-q},\ |q|<1

Sum of all terms of an infinitely decreasing geometric progression (|q| < 1).

Legend:
  • b1 — First term (b1)
  • q — Common ratio (q)

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