How To Calculate and Solve Arithmetic Progression In Sequences and Series

Arithmetic Progression

A sequence is a set of numbers arranged in a definite pattern. Each number is called a term.

A finite sequence is one that has a last term when listed. For Example: 2,4,6,8,,…,16. An infinite sequence is one that does not have a last term when listed.

Arithmetic Progression follows the rule of linear sequence which is the sequence in which each term is obtained by adding a distant number (Positive or Negative) to the proceeding terms.

The constant number is called common difference and it is denoted as “d” while the first term is denoted as “a”.

If T₁, T₂, T₃, T₄, T₅, … is a linear sequence, the common difference is obtained as:

T₂ – T₁ = T₃ – T₂ = T₄ – T₃ = T₅ – T₄
Therefore, d = T_n – T_{(n – 1)}
where
T_n = n^th term of the sequence and T_{(n – 1)} is the (n – 1)^{th</sup< term.}

In other words, here is a quick example:
Lets find the common difference in the following arithmetic progression:
a) 0,5,10,15,20
b) 3¹⁄₄, 5¹⁄₂, 7³⁄₄, 10, 12¹⁄₄

Solution

a) To find the common difference we use the formula, T_n – T_{(n – 1)}
d = T₂ – T₁
T₂ = 5 and T₁ = 0
d = 5 – 0
d = 5
Therefore, the common difference for the first series (a) is 5.

b) Then, by applying the same principle,
d = T₂ – T₁
T₂ = 5^{1 d = 51⁄₂ – 31⁄₄
d = 21⁄₄}

The n^th term of an arithmetic progression is denoted as T_n or U_n,
if T_{1</sub = a
T2 = T1 + d = a + d
T3 = T2 + d = a + d + d = a + 2d}

Therefore, this implies that
T_n = a + (n – 1)d
Hence, this formula above is the general formula used in solving arithmetic progression

Let’s take some examples:

a) Find the n^th term in 34, 40, 46
b) The second, third and fourth terms in an arithmetic progression are x – 2, 5 and x + 2 respectively. Therefore, calculate the value of x.
c) Find the number of terms in an arithmetic progression given that its first and last terms are x and 31x and its common difference is 5x.
d) Find the number of terms and the expression for the n^th term of the following arithmetic progression 32, 29, 26, …, -118
e) The 8^th term of an arithmetic progression is 18 and the 12^th term is 26. So, find the first term, common difference and the 20^th term.
f) The first term of an arithmetic progression is 10, the ratio of the 7^th term to the 9^th term is 3:5. Then, calculate the common difference of the progression.
g) The numbers 11, x, y, 21¹⁄₂ from an arithmetic progression. Hence, find the values of x and y.

Solution

a) We have the formula as T_n = a + (n – 1)d
a = 34
d = T₂ – T₁ = 40 – 34 = 6
Therefore, T_n = 34 + (n – 1)6
T_n = 34 + 6n – 6
T_n = 34 – 6 + 6n
i.e. T_n = 28 + 6n

b)
2^nd term = x – 2
3^rd term = 5
4^th term = x + 2

Since there is no first term, we would use common difference to calculate the value of x

d = T₃ – T₂ = T₄ – T₃
d = 5 – (x – 2) = (x + 2) – 5
5 – x + 2 = x + 2 – 5
5 + 2 – 2 + 5 = x + x
10 = 2x
5 = x
Therefore, the value of x is 5.

c)
a (First Term) = x
T_n = l (Last Term) = 31x
d (Common Difference) = 5x

T_n = a + (n – 1)d
31x = x + (n – 1)5x
31x = x + 5nx – 5x
i.e. 31x = -4x + 5nx
31x + 4x = 5nx
35x = 5nx
7 = n
Therefore, the number of terms is 7.

d)
A.P. => 32, 29, 26, …, -118
So, let’s first of all find the number of terms
General Formula: T_n = a + (n – 1)d
a = 32
d = T₂ – T₁
d = 29 – 32 = -3
l = -118

-118 = 32 + (n – 1)-3
-118 = 32 -3n + 3
So, -118 = 35 – 3n
3n = 35 + 118
3n = 153
n = 51
Therefore, the number of terms is 51.
Then, we find the n^th term expression
T_n = 32 + (n – 1)-3
T_n = 32 + 3 – 3n
i.e. T_n = 35 – 3n

e)
T₈ = 18, n = 8
T₁₂ = 26, n = 12

First, let’s find the T₈ and T₁₂ expression
T₈ = a + (8 – 1)d
T₈ = a + 7d … (I)

T₁₂ = a + (12 – 1)d
T₁₂ = a + 11d … (II)

So, we obtain the values of a and d using simultaneous equation for equation (I) and (II)

Using Elimination Method

18 = a + 7d
-26 = a + 11d

This leaves us with this equation
-8 = -4d
d = -8 / -4 = 2
Common Difference is 2

Using Equation (II) to find the value of a (first term)

26 = a + 11d
That is, 26 = a + 11(2)
26 = a + 22
a = 26 – 22
a = 4
First Term is 4

Ne, we calculate the value of the 20^th term, which will be T₂₀ = 4 + (20 – 1)2
T₂₀ = 4 + 19(2)
T₂₀ = 4 + 38
That is, T₂₀ = 42

f)
First Term (a) = 10
T₇ : T₉ = 3:5 = ³⁄₅

So, by using the general formula: T_n = a + (n – 1)d

T₇ = 10 + (7 – 1)d
T₇ = 10 + 6d

Then,

T₉ = 10 + (9 – 1)d
T₉ = 10 + 8d

Since T₇:T₉ = 3:5
It implies that 5T₇ = 3T₉

5(10 + 6d) = 3(10 + 8d)
50 + 30d = 30 + 24d
30d – 24d = 30 – 50
6d = -20
d = -3.3333

Therefore, the common difference is -3.3333

g)
A.P. => 11, x, y, 21¹⁄₂
General Formula: T_n = a + (n – 1)d
T₄ = 21¹⁄₂, n = 4, a = 11

21¹⁄₂ = 11 + (4 – 1)d
21¹⁄₂ = 11 + 3d
⁴³⁄₂ = 11 + 3d
Multiplying throughout by 2
43 = 22 + 6d
43 – 22 = 6d
21 = 6d
d = 21/6
d = 3.5
Then, Let’s find x

T₂ = x = 11 + (2 – 1)3.5
x = 11 + (1)3.5
x = 11 + 3.5
That is, x = 14.5

So, let’s find y

T₃ = y = 11 + (3 – 1)3.5
y = 11 + (2)3.5
y = 11 + 7
Therefore, y = 18

Series

A series is obtained when the terms of a sequence are added.
For Example:
a) 9 + 12 + 15 + 18 + … + 51
b) 4 + 8 + 12 + .. + 48
c) 1 + 2 + 3 + 4 + …
d) 5 + 10 + 15 + 20 + …

Examples a) and b) are called finite series because they have a definite number of terms and it is always possible to find the sum of a finite series.
Examples c) and d) are called infinite series. The sum of infinite series is often impossible to find.

Let’s solve something
Find the sum of this A.P.
2 + 4 + 6 + … + 98 + 100
Solution
a (First Term) = 2, l (Last Term) = 100
d (Common Difference) = T₂ – T₁
d (Common Difference) = 4 – 2 = 2

T_n = a + (n – 1)d
100 = 2 + (n – 1)2
100 = 2 + 2n – 2
This means 100 = 2n
n = 100/2
n = 50

Therefore, the number of terms in the arithmetic progression is 50

Take a closer look at this

S (Sum) = 2 + 4 + 6 + … + 98 + 100
Reversing the Series
S (Sum) = 100 + 98 + … + 6 + 4 + 2

So, by adding the two series vertically we have:

2S = 102 + 102 + … + 102 + 102 + 102
2S = 102 x 50
Hence, 2S = 5100
S = ⁵¹⁰⁰⁄₂
S = 2550

Therefore, the sum of the arithmetic series is 2550

The following expression represents a general arithmetic series, where the terms are added.

S = a + (a + d) + (a + 2d) + … + l … (III)
S = l + (l – d) + (l – 2d) + … + a … (IV)

So, by adding the equations (III) and (IV)

2S = (a + l) + (a + l) + … + (a + l)
2S = (a + l)n
S = ^{n(a + l)}⁄₂ … (V)

The above formula is used when first term and last term is given

There is another formula

Since l = a + (n – 1)d
Then substituting l = a + (n – 1)d into equation (V)
S_n = ^{n(a + l)}⁄₂
S_n = ^{n(a + a + (n – 1)d)}⁄₂
Therefore, S_n = ^{n(2a + (n – 1)d)}⁄₂ … (VI)

The above equation or equation (VI) is used when the first term and common difference is given.

Let’s take another example

a) The first term and last term of an arithmetic progression are 2 and 256 respectively. Hence, how many terms are there in the sequence and what is the common difference of the series if its sum is 1548?

Solution

a)
First, Let’s find the number of terms, n
a (First Term) = 2
l (Last Term) = 256
S_n (Sum of the Series) = 1548

S_n = ^{n(a + l)}⁄₂
1548 = ^{n(2 + 256)}⁄₂
Then, multiplying throughout by 2
3096 = n(2 + 256)
3096 = 258n
n = ³⁰⁹⁶⁄₂₅₈
n = 12

Therefore, the number of terms of the series is 12

So, let’s find the common difference, d

S_n = ^{n(2a + (n – 1)d)}⁄₂
1548 = ^{12(2(2) + (12 – 1)d)}⁄₂
So, multiplying throughout by 2
This will give you 3096 = 12(4 + (11)d)
3096 = 48 + 132d
3096 – 48 = 132d
3048 = 132d
d = ³⁰⁴⁸⁄₁₃₂
d = 23.0909

Therefore, the common difference is 23.0909.

Nickzom Calculator is capable of solving all the parameters of an Arithmetic Progression.

Download Our Free Android App