Series and Sequences

A sequence is an ordered list of numbers, known as terms. For example, 3, 6, 9, 12... is a sequence. A series is the sum of the terms of a sequence. The corresponding series would be 3 + 6 + 9 + 12 + ... We often use the notation Un to represent the nth term of a sequence and Sn for the sum of the first n terms. This topic explores predictable patterns within sequences and series, specifically Arithmetic and Geometric Progressions, and a powerful tool for expanding expressions called the Binomial Expansion.

### Arithmetic Progressions (AP)

An Arithmetic Progression (AP) is a sequence where the difference between consecutive terms is constant. This constant value is called the common difference (d).

Formula for the nth term of an AP:

To find any term in an AP, we use the formula:

Un = a + (n-1)d

Where:

*Example:* Find the 20th term of the arithmetic sequence 5, 9, 13, 17, ...

Here, a = 5 and d = 9 - 5 = 4.

U20 = 5 + (20-1) * 4 = 5 + 19 * 4 = 5 + 76 = 81.

Formula for the Sum of an AP:

To find the sum of the first n terms of an AP, we use:

Sn = n/2 * [2a + (n-1)d]

An alternative formula, useful when the last term (l) is known, is:

Sn = n/2 * (a + l)

*Example:* Find the sum of the first 15 terms of the sequence 2, 8, 14, ...

Here, a = 2 and d = 8 - 2 = 6.

S15 = 15/2 * [2(2) + (15-1) * 6] = 7.5 * [4 + 14 * 6] = 7.5 * [4 + 84] = 7.5 * 88 = 660.

### Geometric Progressions (GP)

A Geometric Progression (GP) is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r).

Formula for the nth term of a GP:

Un = ar^(n-1)

Where:

*Example:* Find the 7th term of the geometric sequence 3, 6, 12, ...

Here, a = 3 and r = 6/3 = 2.

U7 = 3 * 2^(7-1) = 3 * 2^6 = 3 * 64 = 192.

Formula for the Sum of a GP:

The sum of the first n terms of a GP is given by:

Sn = a(r^n - 1) / (r - 1) or equivalently Sn = a(1 - r^n) / (1 - r). The first is often used when |r| > 1, and the second when |r| < 1 to keep the denominator positive.

*Example:* Find the sum of the first 8 terms of the GP 5, 15, 45, ...

Here, a = 5 and r = 3.

S8 = 5(3^8 - 1) / (3 - 1) = 5(6561 - 1) / 2 = 5(6560) / 2 = 16400.

Sum to Infinity of a GP:

If the common ratio r is between -1 and 1 (i.e., |r| < 1), the terms get progressively smaller, and the sum of the series approaches a finite value. This is called the sum to infinity (S∞).

S∞ = a / (1 - r)

*Example:* Find the sum to infinity of the series 10 + 5 + 2.5 + ...

Here, a = 10 and r = 5/10 = 0.5. Since |0.5| < 1, the sum to infinity exists.

S∞ = 10 / (1 - 0.5) = 10 / 0.5 = 20.

### Binomial Expansion

The Binomial Expansion provides a formula for expanding expressions of the form (a+b)^n where n is a positive integer. The coefficients in the expansion can be found using Pascal's Triangle or the combination formula nCr.

The general formula is:

(a+b)^n = C(n,0)a^n + C(n,1)a^(n-1)b + C(n,2)a^(n-2)b^2 + ... + C(n,n)b^n

Where C(n,r) = n! / (r!(n-r)!).

The general term in the expansion is given by C(n,r) * a^(n-r) * b^r. This is useful for finding a specific term without writing out the full expansion.

*Example:* Find the coefficient of x^3 in the expansion of (2 + x)^5.

We are looking for the term where the power of x is 3. In the general term formula, our 'b' is x, so we need the term where r = 3.

The term is: C(5,3) * 2^(5-3) * x^3.

C(5,3) = 5! / (3!2!) = 10.

The term is 10 * 2^2 * x^3 = 10 * 4 * x^3 = 40x^3.

The coefficient of x^3 is 40.