Arithmetic and Geometric Sequences: Formulas and Sums

An arithmetic sequence adds a constant d. Its formula is ( a_n = a_1 + (n – 1)d ). Its sum is ( S_n = frac{n}{2}(a_1 + a_n) ). A geometric sequence multiplies by a constant r. It follows ( a_n = a_1r^{n-1} ). Its sum is ( S_n = a_1frac{1 – r^n}{1 – r} ). However, if ( lvert rrvert < 1 ), the infinite sum is ( frac{a_1}{1 – r} ). Sigma notation ( sum_{k=1}^{n} ) adds terms from ( k = 1 ) to ( n ).

A sequence is an ordered list of numbers: ( a_1, a_2, a_3, dots ). The two kinds you meet most often are:

• Arithmetic sequences, where you add the same number each time: 3, 7, 11, 15, … (add 4).
• Geometric sequences, where you multiply by the same number each time: 2, 6, 18, 54, … (multiply by 3).

Adding up the terms of a sequence gives a series. This guide covers the formulas for both, sigma notation for writing sums compactly, and when an infinite sum has a finite value, which is where sequences meet calculus.

Arithmetic sequences

An arithmetic sequence has a common difference ( d ), the amount added each step. Find it by subtracting any term from the next: ( d = a_{n+1} – a_n ).

nth term formula

Starting from ( a_1 ), you add ( d ) once to reach ( a_2 ), twice to reach ( a_3 ), and ( n – 1 ) times to reach ( a_n ):

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

Sum formula

The sum of the first n terms is

$$S_n = frac{n}{2}(a_1 + a_n) = frac{n}{2}big(2a_1 + (n - 1)dbig)$$

Why it works. Write the sum forwards and backwards and add the two lines. Each column adds to ( a_1 + a_n ), and there are n columns, so ( 2S_n = n(a_1 + a_n) ). This is the trick a young Carl Friedrich Gauss reportedly used to add 1 to 100 in seconds: ( frac{100}{2}(1 + 100) = 5050 ).

Example 1

For 3, 7, 11, 15, …, find ( a_{20} ) and ( S_{20} ).

Here ( a_1 = 3 ) and ( d = 4 ). Then ( a_{20} = 3 + 19(4) = 79 ), and

$$S_{20} = frac{20}{2}(3 + 79) = 10 cdot 82 = 820$$

Example 2: finding the formula from two terms

An arithmetic sequence has ( a_4 = 17 ) and ( a_{10} = 41 ). Going from term 4 to term 10 adds ( d ) six times, so ( 6d = 24 ) and ( d = 4 ). Then ( a_1 = 17 – 3(4) = 5 ), and ( a_n = 5 + 4(n – 1) = 4n + 1 ).

Geometric sequences

A geometric sequence has a common ratio ( r ), found by dividing any term by the one before: ( r = frac{a_{n+1}}{a_n} ).

nth term formula

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

Sum formula

For ( r neq 1 ),

$$S_n = a_1frac{1 - r^n}{1 - r}$$

Why it works. Multiply ( S_n = a_1 + a_1r + dots + a_1r^{n-1} ) by r. The result ( rS_n = a_1r + dots + a_1r^n ) shares all but two terms with ( S_n ). Subtract: ( S_n – rS_n = a_1 – a_1r^n ), and divide by ( 1 – r ).

Example 3

For 2, 6, 18, 54, …, ( a_1 = 2 ) and ( r = 3 ). The 8th term is ( 2 cdot 3^7 = 4374 ), and

$$S_8 = 2 cdot frac{1 - 3^8}{1 - 3} = 2 cdot frac{-6560}{-2} = 6560$$

Example 4: a shrinking sequence

A ball dropped from 10 m bounces back to 60% of its previous height each time. The heights of the bounces are ( 6, 3.6, 2.16, dots ), a geometric sequence with ( a_1 = 6 ) and ( r = 0.6 ). So, the 5th bounce reaches ( 6(0.6)^4 approx 0.778 ) m.

Infinite geometric series

If ( |r| < 1 ), the powers ( r^n ) shrink toward 0 as n grows, so ( S_n ) approaches a limit:

$$S_infty = sum_{n=1}^{infty} a_1r^{n-1} = frac{a_1}{1 - r}, qquad |r| < 1$$

If ( |r| ge 1 ), the terms don’t shrink to zero and the series diverges.

Example 5: the bouncing ball, totalled

In Example 4, the total distance travelled is the 10 m first drop plus each bounce up and down:

$$10 + 2left( frac{6}{1 - 0.6} right) = 10 + 2(15) = 40 text{ m}$$

An infinite number of bounces covers a finite distance.

Example 6: repeating decimals

( 0.overline{3} = 0.3 + 0.03 + 0.003 + dots ), a geometric series with ( a_1 = 0.3 ) and ( r = 0.1 ), so it equals ( frac{0.3}{0.9} = frac13 ).

Sigma (summation) notation

The Greek capital sigma ( Sigma ) means “add up”. The expression

$$sum_{k=1}^{n} f(k)$$

means: substitute ( k = 1, 2, dots, n ) into ( f(k) ) and add the results. The letter k is the index; 1 and n are the lower and upper limits.

Sigma form	Expanded	Value
( sum_{k=1}^{5} k )	( 1 + 2 + 3 + 4 + 5 )	15
( sum_{k=1}^{4} k^2 )	( 1 + 4 + 9 + 16 )	30
( sum_{k=0}^{3} 2^k )	( 1 + 2 + 4 + 8 )	15
( sum_{k=1}^{10} (4k – 1) )	( 3 + 7 + dots + 39 )	210

Useful rules: constants factor out, ( sum c,a_k = csum a_k ), and sums split, ( sum (a_k + b_k) = sum a_k + sum b_k ). As a result, three standard results come up constantly in calculus:

$$sum_{k=1}^{n} 1 = n, qquad sum_{k=1}^{n} k = frac{n(n+1)}{2}, qquad sum_{k=1}^{n} k^2 = frac{n(n+1)(2n+1)}{6}$$

These are exactly what you need to evaluate Riemann sums by hand and see an integral appear as a limit.

Sequence calculator

Choose arithmetic or geometric, then enter the first term, the common difference or ratio, and n. Switch to geometric with ( r = 0.5 ) to see an infinite sum.

The full-page sequence calculator has more examples. Also, for series that aren’t geometric, the series convergence calculator sums them numerically.

Sequences in calculus

In Calculus 2, sequences and series become a major unit:

• A sequence converges if ( lim_{ntoinfty} a_n ) exists. For example ( frac{n}{n+1} to 1 ). See limits at infinity.
• The geometric series formula is the benchmark that most convergence tests compare against.
• The ratio test checks whether a series behaves like a geometric series with ( |r| < 1 ).
• Taylor series write functions such as ( e^x ) and ( sin x ) as infinite sums.

Common mistakes

• Using n instead of n − 1. ( a_1 ) has no added d and no factor of r, so the exponent and multiplier are ( n – 1 ).
• Confusing d and r. Differences are constant in arithmetic sequences; however, ratios are constant in geometric ones. Check two pairs of terms to be sure.
• Applying ( frac{a_1}{1-r} ) when ( |r| ge 1 ). For 1, 2, 4, 8, … the formula would give ( -1 ), which is nonsense; the series diverges.
• Miscounting terms. ( sum_{k=0}^{10} ) has 11 terms, not 10.

Practice problems

1. Find the 15th term of 7, 12, 17, 22, …
2. Find the sum of the first 30 terms of the arithmetic sequence with ( a_1 = 2 ) and ( d = 3 ).
3. Find the 6th term of the geometric sequence 5, 10, 20, …
4. Evaluate ( sum_{k=1}^{6} 3 cdot 2^{k-1} ).
5. Find the sum of the infinite series ( 12 + 4 + frac43 + dots )
6. Write ( 0.overline{45} ) as a fraction.

Answers: (1) 77; (2) 1365; (3) 160; (4) 189; (5) 18; (6) ( frac{5}{11} ).

FAQ

What is the arithmetic sequence formula?

( a_n = a_1 + (n – 1)d ), where ( a_1 ) is the first term and d is the common difference. The sum of the first n terms is ( frac{n}{2}(a_1 + a_n) ).

What is the geometric sequence formula?

( a_n = a_1r^{n-1} ), with sum ( S_n = a_1frac{1 – r^n}{1 – r} ). The infinite sum is ( frac{a_1}{1 – r} ) when ( |r| < 1 ).

What is the difference between a sequence and a series?

A sequence is a list of terms. A series, however, is the sum of those terms.

What does the sigma symbol mean in math?

( Sigma ) means summation: add the expression for every value of the index between the lower and upper limits.

Can a sequence be both arithmetic and geometric?

Only a constant sequence, like 5, 5, 5, …, which has ( d = 0 ) and ( r = 1 ).

Further reading

• Sequences (Paul’s Online Math Notes) — sequence notation, limits of sequences and convergence.
• Summation Notation (Paul’s Online Math Notes) — sigma notation, its properties and the standard sum formulas.
• Geometric series (Wikipedia) — history, proofs of the sum formula and applications.