Maths and Computing

nth term

First term:

a

Common difference:

d

Sequence:

a a+d a+2d a+3d ...

nth term:

a + (n - 1)d

Iterative definition:

u₁ = a, u_n = u_n-1 + d for all n ≥ 2

Sum of first n terms:

n(2a + (n - 1)d)

2

a

d

n

nth term

First term:

a

Common difference:

r

Sequence:

a ar ar² ar³ ...

nth term:

ar^n-1

Iterative definition:

u₁ = a, u_n = u_n-1 × r for all n ≥ 2

Sum of first n terms:

a(rn-1)

r - 1

Series behaviour:

The series diverges for |r| ≥ 1 and converges for |r| < 1

Sum of infinite terms:

a

1 - r

a

r

n

nth term

First term:

1

nth term:

n^r

Sequence:

1^r 2^r 3^r 4^r 5^r 6^r ...

r

n

nth term

Period:

p

1

Initial terms:

u₁

Sequence posision:

k where k = ((n-1) mod p) + 1

nth term:

u_n = u_k

Iterative definition:

u₁ u₂, ... u_p given. u_n = u_{n - p} for n > p

Sum of first n terms:

(u₁ + ... + u_p) × q + (u₁ + ... + u_k) where q = n/p rounded down

p

n

nth term

First term:

1

nth term:

n(n+1)

2

(n-1)th term:

n(n-1)

2

Term difference:

n

Iterative definition:

u₁ = 1, u_n = u_n-1 + n for all n ≥ 2

Sequence:

1 3 6 10 15 21 ...

Sum of first n terms:

n(n+1)(n+2)

6

Series behaviour:

The series diverges

n

nth term

First term:

1

nth term:

n! (n × (n-1) × (n-2) × ... × 1)

(n-1)th term:

(n-1)!

Iterative definition:

u₁ = 1, u_n = u_n-1 × n for all n ≥ 2

Sequence:

1 2 6 24 120 720 ...

n

nth term

First two terms:

1, 1

nth term:

n(n+1)

2

nth term

1 √5

((

1 + √5 2

)n

- (

1 - √5 2

)n)

(n-1)th term:

(n-2)nd term:

Term difference:

(n-2)th term

Iterative definition:

u₁ = 1, u₂ = 1, u_n = u_n-1 + u_n-2 for all n ≥ 3

Sequence:

1 1 2 3 5 8 13 22 ...

Sum of first n terms:

1 √5

((

1 + √5 2

)n+2

- (

1 - √5 2

)n+2) - 1 or un+2 - 1

Series behaviour:

The series diverges

n

nth term

First term:

2

nth term:

There is no formula for the nth prime

(n-1)th term:

Sequence:

2 3 5 7 11 13 17 19 ...

Number of terms:

There are infinitely many prime numbers

n

nth term

First term:

1

nth term:

n

(n-1)th term:

n - 1

Term difference:

1

Sequence type:

Algebraic: a = 1, d = 1

Iterative definition:

u₁ = 1, u_n = u_n-1 + 1 for all n ≥ 2

Sequence:

1 2 3 4 5 6 ...

Sum of first n terms:

n(n + 1)

2

n

nth term

First term:

1

nth term:

n²

(n-1)th term:

(n - 1)²

Term difference:

2n - 1 (nth odd number)

Sequence type:

Power: r = 2

Iterative definition:

u₁ = 1, u_n = u_n-1 + 2n - 1 for all n ≥ 2

Sequence:

1 4 9 16 25 36 ...

Sum of first n terms:

n(n + 1)(2n + 1)

6

n

nth term

First term:

1

nth term:

n³

(n-1)th term:

(n - 1)³

Term difference:

3n² - 3n + 1

Sequence type:

Power: r = 3

Iterative definition:

u₁ = 1, u_n = u_n-1 + 2n - 1 for all n ≥ 2

Sequence:

1 8 27 64 125 216 ...

Sum of first n terms:

n2(n+1)2

4

n

nth term

First term:

1

nth term:

2^n-1

(n-1)th term:

2^n-2

Sequence type:

Geometric: a = 1, r = 2

Iterative definition:

u₁ = 1, u_n = u_n-1 × 2 for all n ≥ 2

Sequence:

1 2 4 8 16 32 64 128 ...

Sum of first n terms:

2ⁿ - 1

n

nth term

First term:

1

nth term:

2n - 1

(n-1)th term:

2n - 3

Term difference:

2

Sequence type:

Algebraic: a = 1, d = 2

Iterative definition:

u₁ = 1, u_n = u_n-1 + 2 for all n ≥ 2

Sequence:

1 3 5 7 9 11 ...

Sum of first n terms:

n²

n

nth term

First term:

2

nth term:

2n

(n-1)th term:

2n - 2

Term difference:

2

Sequence type:

Algebraic: a = 2, d = 2

Iterative definition:

u₁ = 2, u_n = u_n-1 + 2 for all n ≥ 2

Sequence:

2 4 6 8 10 12 ...

Sum of first n terms:

n(n + 1)

n