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Arithmetic and Geometric Sequences and their Summation

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### 14.1 Sequences

### Consider the following sequence:1, 3, 5, 7, 9, ….., 111

### Consider the sequence 2, 4, 8, 16, ….

### Arithmetic Means

### Geometric Means

### Formula of Arithmetic Series

### S(n) = a + aR + aR2 +aR3+ …. + aRn-1

### Consider such a Geometric Series

arithmetic sequence

geometric sequence

geometric sequence

geometric sequence

Find the next two terms of the following sequences :

2, 5, 8, 11,……

2, 6, 18, 54, ….

2, 4, 8, 16,…….

5, -25, 125, -625, ….

3, 4, 6, 9, 13, …….

5, 2, -1, -4, …..

0, sin20o, 2sin30o, 3sin40o

arithmetic sequence

3 is the second term of the sequence, mathematically,

T(2) = 3 or T2 = 3

1 is the first term of the sequence,mathematically, T(1) = 1 or T1 = 1

5 is the third term of the sequence, mathematically, T(3) = 5 or T3 = 5

111 is the nth term of the sequence, mathematically, T(n) = 111 or Tn = 111

So, the sequence can be represented by the general term

T(n) = 2n or Tn = 2n

The sequence is formed from timing 2 to the previous term.

Illustrative Examples

When a, b and c are three consecutive terms of arithmetic sequence, the middle term b is called the arithmetic mean of a and c.

Insert two arithmetic means between 11 and 35.

When x, y and z are three consecutive terms of geometric sequence, the middle term y is called the geometric mean of x and z.

Insert two geometric means between 16 and -54.

The expression T(1) + T(2) + T(3) +….+ T(n) is called a series. We usually denote the sum of the first n term of a series by the notation S(n).

Let’s consider a sequence : T(1), T(2), T(3), T(4), …., T(n)

S(n) = l + l - d + l - 2d + l - 3d + …. + a + d+ a

S(n) = a + a + d + a + 2d + a + 3d + ………... + a + (n - 1)d

S(n) = l + l – d + l - 2d + l - 3d + ….+ a + d+ a

2S(n) =(a + l)+(a + l)+(a + l)+(a + l)+….. +(a + l)

2S(n) = n(a + l)

S(n) –R.S(n) = a - aRn

R.S(n) = aR + aR2 + aR3 +aR4+ …. + aRn

(1 – R) S(n) = a (1 – Rn)

Timing –1 on both numerator and denominator

Sum to Infinity of a Geometric Series

What is the value of common ratio R ?

What will occur for if n tends to infinity ?

where –1< R <1

Summation Notation

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