UNIT 7 • ALGEBRA

AP & GP
nth Term and Sum of a Series

An arithmetic progression adds a fixed number d each time: $a_n = a + (n - 1)d$ . A geometric progression multiplies by a fixed number r: $a_n = ar^{n-1}$ . Each has a formula for the sum of its first n terms.

THE FORMULASAP AND GP

\begin{gathered} \text{AP: } a_n = a + (n - 1)d, \quad S_n = \frac{n}{2}\left[2a + (n - 1)d\right] \\ \text{GP: } a_n = ar^{n - 1}, \quad S_n = \frac{a(r^n - 1)}{r - 1}, \quad S_\infty = \frac{a}{1 - r} \ (|r| < 1) \end{gathered}

1. AP or GP?Subtract neighbours: constant difference means AP. Divide them: constant ratio means GP.

2. Read a and d (or r)a is the first term; d = second − first; r = second ÷ first.

3. SubstitutePut a, d (or r) and n into the formula. Count n carefully.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

For the AP 5, 9, 13, 17, …, find the 20th term and the sum of the first 20 terms.

STEP 01 • FIRST TERM AND COMMON DIFFERENCE

a = 5, \qquad d = 9 - 5 = 4

STEP 02 • THE 20TH TERM

a_{20} = a + 19d = 5 + 19 \times 4 = 81

STEP 03 • SUM OF 20 TERMS

S_{20} = \frac{20}{2}\left[2(5) + 19(4)\right] = 10 \times 86 = 860

Check with the first and last terms: $\frac{20}{2}(5 + 81) = 860$ ✓

FINAL CONCLUSION

a₂₀ = 81, S₂₀ = 860

a_{20} = 81, \qquad S_{20} = 860

The calculator below opens on this AP. Switch to GP or "AP or GP?" to test a sequence of your own.

LABORATORY

Interactive AP & GP Calculator

LABORATORY

Progressions Engine

What do you want to work out?

First term aCommon difference dNumber of terms n

Common Questions

What is the difference between an AP and a GP?

In an arithmetic progression you add the same number each time (2, 5, 8, 11: d = 3). In a geometric progression you multiply by the same number (3, 6, 12, 24: r = 2).

What is the formula for the sum of an AP?

S_n = \frac{n}{2}\left[2a + (n - 1)d\right]

, or

\frac{n}{2}(a + l)

when you know the last term l. It comes from pairing the first and last terms, as Gauss did with 1 + 2 + … + 100 = 5050.

When does a GP have a sum to infinity?

Only when

|r| < 1

, so the terms shrink towards 0. Then

S_\infty = \frac{a}{1 - r}

: for 1 + ½ + ¼ + … it is 2.

What are Σn, Σn² and Σn³?

The sums of the first n natural numbers, their squares and their cubes:

\frac{n(n+1)}{2}

\frac{n(n+1)(2n+1)}{6}

and

\left[\frac{n(n+1)}{2}\right]^2

PreviousSystem of Equations

Unit 7•49 of 63

NextPermutations & Combinations

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 7 • ALGEBRA

AP & GP
nth Term and Sum of a Series

THE FORMULASAP AND GP

\begin{gathered} \text{AP: } a_n = a + (n - 1)d, \quad S_n = \frac{n}{2}\left[2a + (n - 1)d\right] \\ \text{GP: } a_n = ar^{n - 1}, \quad S_n = \frac{a(r^n - 1)}{r - 1}, \quad S_\infty = \frac{a}{1 - r} \ (|r| < 1) \end{gathered}

1. AP or GP?Subtract neighbours: constant difference means AP. Divide them: constant ratio means GP.

2. Read a and d (or r)a is the first term; d = second − first; r = second ÷ first.

3. SubstitutePut a, d (or r) and n into the formula. Count n carefully.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

For the AP 5, 9, 13, 17, …, find the 20th term and the sum of the first 20 terms.

STEP 01 • FIRST TERM AND COMMON DIFFERENCE

a = 5, \qquad d = 9 - 5 = 4

STEP 02 • THE 20TH TERM

a_{20} = a + 19d = 5 + 19 \times 4 = 81

STEP 03 • SUM OF 20 TERMS

S_{20} = \frac{20}{2}\left[2(5) + 19(4)\right] = 10 \times 86 = 860

Check with the first and last terms: $\frac{20}{2}(5 + 81) = 860$ ✓

FINAL CONCLUSION

a₂₀ = 81, S₂₀ = 860

a_{20} = 81, \qquad S_{20} = 860

The calculator below opens on this AP. Switch to GP or "AP or GP?" to test a sequence of your own.

LABORATORY

Interactive AP & GP Calculator

LABORATORY

Progressions Engine

What do you want to work out?

First term aCommon difference dNumber of terms n

Common Questions

What is the difference between an AP and a GP?

In an arithmetic progression you add the same number each time (2, 5, 8, 11: d = 3). In a geometric progression you multiply by the same number (3, 6, 12, 24: r = 2).

What is the formula for the sum of an AP?

S_n = \frac{n}{2}\left[2a + (n - 1)d\right]

, or

\frac{n}{2}(a + l)

when you know the last term l. It comes from pairing the first and last terms, as Gauss did with 1 + 2 + … + 100 = 5050.

When does a GP have a sum to infinity?

Only when

|r| < 1

, so the terms shrink towards 0. Then

S_\infty = \frac{a}{1 - r}

: for 1 + ½ + ¼ + … it is 2.

What are Σn, Σn² and Σn³?

The sums of the first n natural numbers, their squares and their cubes:

\frac{n(n+1)}{2}

\frac{n(n+1)(2n+1)}{6}

and

\left[\frac{n(n+1)}{2}\right]^2

PreviousSystem of Equations

Unit 7•49 of 63

NextPermutations & Combinations

AP & GP nth Term and Sum of a Series

Step-by-Step Worked Example

a₂₀ = 81, S₂₀ = 860

Interactive AP & GP Calculator

Progressions Engine

Common Questions

NETZ Engine

AP & GP nth Term and Sum of a Series

Step-by-Step Worked Example

a₂₀ = 81, S₂₀ = 860

Interactive AP & GP Calculator

Progressions Engine

Common Questions

AP & GP
nth Term and Sum of a Series

AP & GP
nth Term and Sum of a Series