Sample Problems on Arithmetic Progressions

Problem 1. Find the sum of the first 35 terms of series 5,11,17,23…..

Solution:

In the given series,

a = 5, d = a₂ – a₁ = 11 – 5 = 6, n = 35

S_n = (n/2)(2a + (n – 1) x d)

S_n = (35/2)(2 x 5 + (35 – 1) x 6)

S_n = (35/2)(10 + 34 x 6)

S_n = (35/2)(10 + 204)

S_n = 35 x 214/2

S_n = 3745

Problem 2. Find the sum of the series when the first term of the series is 5 and the last of the series is 209 and the number of terms in the series is 35.

Solution:

In the given series,

a = 5, l = 209, n = 35

S_n = (n/2)(a + l)

S_n = (35/2)(5 + 209)

S_n = 35 x 214/2

S_n = 3745

Problem 3. 21 Rupees is divided among three brothers where the three parts of money are in AP and the sum of their squares is 155. Find the largest amount.

Solution:

Let the money is deivided in three parts as (a-d), a, (a+d)

Given,

(a – d) + a + (a + d) = 21

Therefore, 

3a = 21

a = 7

Again, (a – d)² + a² + (a + d)² = 155

a² + d² – 2ad + a² + a² + d² + 2ad = 155

3a² + 2d² = 155

Putting the value of ‘a’ we get,

3(7)² + 2d² = 155

2d² = 155 – 147

d² = 4

d = ±2

Three parts of distributed money are:

a + d = 7 + 2 = 9

a = 7

a – d = 7 – 2 = 5

Hence, the largest part is Rupees 9

Arithmetic progression(AP) also called an arithmetic sequence is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence

A progression is a sequence or series of numbers in which they are arranged in a particular order such that the relation between the consecutive terms of a series or sequence is always constant. In a progression, it is possible to obtain the n_th term of the series.

In mathematics, there are 3 types of progressions:

1. Arithmetic Progression (AP)
2. Geometric Progression (GP)
3. Harmonic Progression (HP)

let’s learn about AP in this article.