Algebra

Arithmetic vs Geometric Sequences: Formulas and Examples

By Math Solving Space · · 2 min read

On this page
  1. How to tell them apart
  2. Arithmetic sequences
  3. Geometric sequences
  4. Sum to infinity
  5. Side by side
  6. Common mistakes
  7. Practice questions
  8. Frequently asked questions

An arithmetic sequence adds the same number each time (3,7,11,15,…3, 7, 11, 15, \ldots). A geometric sequence multiplies by the same number each time (2,6,18,54,…2, 6, 18, 54, \ldots). Arithmetic sequences grow in a straight line; geometric ones grow (or shrink) exponentially.

How to tell them apart

Check the gaps between terms:

  • Same difference each time → arithmetic. The common difference is dd.
  • Same ratio each time → geometric. The common ratio is rr.

5,8,11,145, 8, 11, 14: differences are all 33, so arithmetic. 3,6,12,243, 6, 12, 24: ratios are all 22, so geometric.

Arithmetic sequences

an=a1+(n−1)dSn=n2(a1+an)a_n = a_1 + (n - 1)d \qquad S_n = \frac{n}{2}(a_1 + a_n)

Example: 3,7,11,…3, 7, 11, \ldots with a1=3a_1 = 3, d=4d = 4. The 10th term and the sum of the first 10 terms:

a10=3+9×4=39S10=102(3+39)=210a_{10} = 3 + 9 \times 4 = 39 \qquad S_{10} = \frac{10}{2}(3 + 39) = 210

Geometric sequences

an=a1rn−1Sn=a1(1−rn)1−r(r≠1)a_n = a_1 r^{n-1} \qquad S_n = \frac{a_1(1 - r^n)}{1 - r} \quad (r \ne 1)

Example: 2,6,18,…2, 6, 18, \ldots with a1=2a_1 = 2, r=3r = 3:

a8=2×37=4374S8=2(1−38)1−3=6560a_8 = 2 \times 3^7 = 4374 \qquad S_8 = \frac{2(1 - 3^8)}{1 - 3} = 6560

Sum to infinity

If −1<r<1-1 < r < 1, the terms shrink towards zero and the total settles at

S∞=a11−rS_\infty = \frac{a_1}{1 - r}

Example: 1+12+14+⋯1 + \frac{1}{2} + \frac{1}{4} + \cdots has S∞=11−12=2S_\infty = \frac{1}{1 - \frac{1}{2}} = 2. After 10 terms the sum is already 1023512≈1.998\frac{1023}{512} \approx 1.998.

If ∣r∣≥1|r| \ge 1 the terms do not shrink, so there is no sum to infinity.

Side by side

Arithmetic Geometric
Rule Add dd Multiply by rr
nth term a1+(n−1)da_1 + (n-1)d a1rn−1a_1 r^{n-1}
Sum of n terms n2(a1+an)\frac{n}{2}(a_1 + a_n) a1(1−rn)1−r\frac{a_1(1-r^n)}{1-r}
Graph of terms Straight line Exponential curve
Real-life example Saving the same amount weekly Compound interest

Common mistakes

  • Using nn instead of n−1n - 1 in the nth-term formulas.
  • Assuming every sequence is arithmetic. Always check the ratio too.
  • Using the sum to infinity when ∣r∣≥1|r| \ge 1.

Practice questions

  1. Find the 20th term of 100,95,90,…100, 95, 90, \ldots
  2. Find the 6th term of 5,−10,20,…5, -10, 20, \ldots
  3. Find the sum to infinity of 9+3+1+⋯9 + 3 + 1 + \cdots

Answers: 1) 55 2) −160-160 3) 272\frac{27}{2}

Try the arithmetic sequence calculator and the geometric sequence calculator, which charts the partial sums approaching the limit. Geometric growth is also the maths behind compound interest.

Frequently asked questions

Can a sequence be both arithmetic and geometric?

Only a constant sequence like 4, 4, 4, …, where d = 0 and r = 1.

What is the difference between a sequence and a series?

A sequence is a list of terms. A series is the sum of the terms.

How do I find the common ratio?

Divide any term by the term before it.

Open the calculator →

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