Arithmetic Progressions are studied under Sequences and series. There are different types of series like the Geometric progressions also. In modular arithmetic , Arithmetic Progressions are very important , Let's few basics about the math progressions and Arithmetic Progressions.
- Look at the list of numbers
1, 3, 5, 7………
- Each of the numbers in the list is called a term.
- An arithmetic progression can be explained as numbers list in every term and is obtained by adding fixed number to the preceding term except the first term.
- This fixed number is called the common difference of the Arithmetic Progression.
It can be positive, negative or zero
- AP's general form is:
a, a+d, a+2d, a+3d, . . .
- An Arithmetic Progressions with finite number of terms is a finite Arithmetic Progression. That means the Arithmetic Progression has a last term.
- An Arithmetic Progressions which does not have finite number of terms is an infinite Arithmetic Progression. That means the Arithmetic Progression does not have a last term.
nth term of Arithmetic Progressions :
Let a1, a2, a3, . . . be an Arithmetic Progressions whose first term a1 is a and the common difference is d.
Then,
the second term a2 = a + d = a + (2 - 1)d
the third term a3 = a2 + d = (a + d) + d = a + 2d = a + (3 - 1) d
the forth term a4 = a3 + d = (a + 2d) + d = a + 3d = a + (4 - 1) d
. . . . . . . . . .
. . . . . . . . .
Looking at the pattern, we can say that the nth term an = a + (n -1) d.
So the nth term an of the Arithmetic Progression with first term a and common difference d is given by an = a + (n -1) d.
The sum of first n terms of an Arithmetic Progression is given by
s = $\frac{n}{2}$ [2a + (n - 1) d]
Arithmetic Progression Formula
The following algebra homework help problems explains the progressions.
Solved Example
Question: Find the sum of the first 22 terms of the Arithmetic Progression: 8, 3, -2, . . .
Solution:
Here, a = 8, d = 3 - 8 = -5 and n = 22
s = $\frac{n}{2}$ [2a + (n - 1) d]
s22 = $\frac{22}{2}$ [16 + (21) (-5)]
= 11(16 - 105)
= 11(-89)
= -979
Similar problems we can see in geometric progressions as well.
Solution:
Here, a = 8, d = 3 - 8 = -5 and n = 22
s = $\frac{n}{2}$ [2a + (n - 1) d]
s22 = $\frac{22}{2}$ [16 + (21) (-5)]
= 11(16 - 105)
= 11(-89)
= -979
Similar problems we can see in geometric progressions as well.
Arithmetic Progression Problems
Below you could see arithmetic progression problems
Solved Examples
Question 1: Find the 10th term of the Arithmetic Progression : 2, 7, 12,. . .
Solution:
Here, a = 2, d = 7 - 2 = 5 and n = 10.
We have an = a + (n - 1) d
So, a10 = 2 + (10 - 1) x 5 = 2 + 45 = 47
Therefore, the 10th term of the given Arithmetic Progression is 47.
Solution:
Here, a = 2, d = 7 - 2 = 5 and n = 10.
We have an = a + (n - 1) d
So, a10 = 2 + (10 - 1) x 5 = 2 + 45 = 47
Therefore, the 10th term of the given Arithmetic Progression is 47.
Question 2: Identify whether the following terms form an Arithmetic Progressions , if they form an Arithmetic Progression, find the common difference and write three more
terms. 2, $\frac{5}{2}$, 3, $\frac{7}{2}$……….
Solution:
The next three terms are 4, $\frac{9}{2}$, 5
terms. 2, $\frac{5}{2}$, 3, $\frac{7}{2}$……….
Solution:
The next three terms are 4, $\frac{9}{2}$, 5
Question 3: Identify whether the following terms form an A.P if they form an Arithmetic Progression, find the common difference and write three more terms. -1.2, -3.2, -5.2, -7.2
Solution:
the next three terms are -9.2, -11.2 and -13.2.
Solution:
the next three terms are -9.2, -11.2 and -13.2.
Question 4: Which term of the AP 21, 18, 15,……. is -81……. Also. Is any term 0? Give reason for your answer.
Solution:
The nth term is 8
Solution:
The nth term is 8
Question 5: Determine the AP where 3rd term is 5 and 7th term is 9
Solution:
The Arithmetic Progressions is 3, 4, 5,6,……………
Solution:
The Arithmetic Progressions is 3, 4, 5,6,……………
Question 6: How many two digit numbers are divisible by 3 the list of two digit numbers divisible by 3 are 12, 15, 18, …………………..99
Solution:
There are 30, 2-digit number divisible by 3
Solution:
There are 30, 2-digit number divisible by 3
Question 7: The sum of 4th and 8th terms of an AP is 24 and the sum of 6th
and 10th terms is 44. Find the first three terms of the AP.
Solution:
the first three terms of the AP are -13, -8, -3,……
and 10th terms is 44. Find the first three terms of the AP.
Solution:
the first three terms of the AP are -13, -8, -3,……
Question 8: How many terms of the AP must be taken so that their sum is 78? The SP is 24, 21, 18,…...
Solution:
The number of terms is either 13 or 4
Solution:
The number of terms is either 13 or 4
Question 9: Find the sum of first 1000 positive integers
Solution:
500500
Solution:
500500
Question 10: A manufacture of TV sets produced 600 sets in the third year and 700 set in the seventh year. Assumes that the production increases uniformly every year. Find
a) Production in the first year
b) Production in the tenth year
c) Total production in first seven years
Solution:
a) Production in first year = 550
b)Production in the tenth year=775
c) Total production in first seven years=4375.
a) Production in the first year
b) Production in the tenth year
c) Total production in first seven years
Solution:
a) Production in first year = 550
b)Production in the tenth year=775
c) Total production in first seven years=4375.
Question 11: In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class in which they are studying, eg. A section of class I will plant 1 tree, a section of class II will plant 2 trees and so on till class XII. There are three sections in each class. How many trees will be planted by the students.
Solution:
234
Solution:
234
Question 12: A sum of Rs. 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs. 20 less than its preceding prize, find the value of the prizes.
Solution:
The prize values are Rs 160, Rs 140, Rs 120, Rs 100, Rs. 80, Rs. 60 Rs. 40.
Solution:
The prize values are Rs 160, Rs 140, Rs 120, Rs 100, Rs. 80, Rs. 60 Rs. 40.
Question 13: A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows. Rs 200 for first day, Rs. 250 for the second day. Rs. 300 for the third day etc, the penalty for each succeeding day being Rs. 50 more than for the preceding day. How much money the contractor has to pay as penalty, if he has delayed the work by 30 days.
Solution:
Rs.27750
Solution:
Rs.27750
Question 14: In an AP, the first term is 8, nth term is 33 and sum to n terms is 123. Find n and d, the common difference.
Solution:
n=6, d=5
Solution:
n=6, d=5
Question 15: For what value of ‘n’ are the nth terms of two APs 63, 65, 67……. and 3, 10, 17… are equal.
Solution:
For n=13, the terms of both the series are equal.
Solution:
For n=13, the terms of both the series are equal.
Question 16: The sum of the 4th and 8th terms of an Arithmetic Progressions is 24 and the sum of 6th and 10th terms if 44. Find the first three terms of the AP.
Solution:
The first 3 terms are -13, -8, -3,…..
Solution:
The first 3 terms are -13, -8, -3,…..