Arthimatic progression

The sum of an arithmetic series

• Let us write the series down again, but this time we shall write it down with the terms in reverse order.
• We get Sn = ℓ + (ℓ − d) + (ℓ − 2d) + . . . + (a + 2d) + (a + d) + a
• 2Sn = (a + ℓ) + (a + ℓ) + (a + ℓ) + . . . + (a + ℓ) + (a + ℓ) + (a + ℓ),
• 2Sn = n(a + ℓ)

The sum of an arithmetic series

• Sn = ½ n(a + ℓ).
• ℓ = a + (n − 1)d ,
• Sn = ½ n(a + ℓ)
• Sn = ½ n(a + a + (n − 1)d)
• Sn= ½ n(2a + (n − 1)d).

The sum of an arithmetic series

• Example -1: Find the sum of the first 50 terms of the sequence 1, 3, 5, 7, 9, . . . .

Solution : This is an arithmetic progression,

a = 1 , d = 2 , n = 50 .

Sn = ½ n(2a + (n − 1)d)

S50 = ½ × 50 × (2 × 1 + (50 − 1) × 2)

= 25 × (2 + 49 × 2)

= 25 × (2 + 98) = 2500

The sum of an arithmetic series

• Example-2: Find the sum of the series 1 + 3·5 + 6 + 8·5 + . . . + 101 .

Solution : Given a=1 , d= 3.5 - 1 = 2.5 and ℓ=101

we know ℓ = a + (n − 1)d

101 = 1 + (n − 1) × 2·5 .

100 = (n − 1) × 2·5

40 = n − 1

n = 41

So, Sn = ½ n(a+ ℓ)= ½ ×41(1+101)=2091

Example 4: Find number of terms of

A.P. 100, 105, 110, 115,,………500

Solution : First term is a = 100 , an = 500

Common difference is d = 105 -100 = 5

nth term is

an = a + (n-1)d

500 = 100 + (n-1)5

500 - 100 = 5(n – 1)

400 = 5(n – 1)

5(n – 1) = 400

5(n – 1) = 400 n – 1 = 400/5 n - 1 = 80

n = 80 + 1

n = 81

Hence the no. of term is 81.

MIND MAP

sn=n(a+l)

2

If a, b, c, are in AP,

2

b is arithmetic mean

b = a + c

Arithmetic Progression

When first term of common differnce is given :

When first & last terms are given:

From beginning an = a+(n–1)d Here

d

–

co

m

on

a – first term

di

2

or sn =

n(n+l)

3

n = 30

i.e., n–1 = 87 = 29

2-digit numbers divisible by 3 12, 15, 18, ... 99

a = 12, d = 3, an = 99

an = a + (n–1)d 99 = 12 + (n–1)3

sn=n(a+l) =n(1+n)

2 2

Let sn = 1 + 2 + 3 + ... n

a = 1, last term l = n

Sum of first n positive integers

How many 2-digit numbers are divisible by 3?

Fixed number in arithmetic progression which provides the to and fro terms by adding/ subtracting from the present number.

Can be positive or negative.

a, a+d, a+2d, a+3d, ... a+(n –1) d

List of numbers in which each term is obtained by adding a fixed number

to the preceding term except the first term. Fixed number is called common difference.

er

h Te

𝑛

𝑆𝑛 = 2 {2𝑎 + (𝑛 − 1)𝑑}

𝑛

𝑆𝑛 = 2 (𝑎 + 𝑎𝑛)

a- First term d- common difference