Arithmetic progression
Is a series of numbers in which the next number can be found by adding a common number to previous number.
If we know first term and common difference we can easily find the next number.
Example: 3,6,9,....... find next term
Hear first term is 3 and common difference is 6-3=3
a4 =a+3d
=3+3×3
=3+9
=12
Therefore here the fourth term is 12
In arithmetic progression
First term is denoted by : a1
Second term is denoted by :a2
Third term is denoted by :a3
Like so on
And last term is denoted by : an.
The common difference is denoted by 'd'
Common difference can be found by the formula.
d = next term - previous term of A.P
d=a2-a1
Common difference also can be found by the formula
d= (ap-aq)/(p-q)
NOTE:- in A.P common difference between all pair of consecutive terms must equal .
Example:- check whether the give number of terms are in A.P
5,8,11,14......
Sol: here a1=5,
a2=8
a3=11
a4=14
To check are they in A.P we find d.
d =a2-a1
d=8-5
d=3
Again,
d=a3-a2
d=11-8
d=3
d=a4-a3
d=14-11
d=3
Therefore
Here common difference is same between all terms,hence it is A.P
Find next term of A.P
1.find next term of given A.P 5,8,11,14...
Sol. d=a2-a1. a4=14. a5=?
=8-5
=3
Here we know that
a5 = a4 + d
a5 = 14 + 3
a5 = 17
2. find 12 th term if 11 term is 121 and common difference is 11.
Sol. here a11 = 121 and d = 11 ,a12=?
a12 = a11+ d
a12 = 121 + 11
a12 = 132
Like this we can find next term if previous term and common difference is given.
Find nth term if first term and common difference given
Here it give first term a and common difference d to find nth term an.
1. Find 22nd term if first term is 10 and common difference is 5.
Sol. Here a=10 and d=5 ,a22=?
For finding nth term we have formula
an = a + (n-1)d
a22 = 10 + (22 - 1)5
a22 = 10 + (11)5
a22 = 10 + 55
a22 = 65
Therefore 22th term is 65
2. Find 100 th term if first term is 5 and common difference is 7.
Sol. a=5 and d=7 ,a100 =?
an = a + (n-1)d
a100 = 5 + (100-1)7
a100 = 5 +(99)7
a100 = 5 + 693
a100 = 698
Therefore 100th term is 698.
3. Find 7th term if first term is -7 and common difference is-9.
Sol. here a = -7 and d = -9 a7 = ?
Sol. an = a + (n-1)d
a7 = -7 + (7-1)(-9)
a7 = -7 +(6)(-9)
a7 = -7-54
a7 =-61
Finding number of terms if first term common difference and last term is given.
1. Find which term of A.p is 120 if first term is 12 and common difference is 2.
Sol. a=12 d=2 an=120 n=?
an=a+(n-1)d
120=12+(n-1)2
120-12=(n-1)2
108=(n-1)2
n-1 =108/2
n-1=54
n=54+1
n=55
2. Find which term of A.p is 340 if first term is 8 and common difference is 4.
Sol. a=8 d=4 an=340 n=?
an=a+(n-1)d
340=8+(n-1)4
340-8= (n-1)4
332=(n-1)4
(n-1)=332/4
n-1=83
n=83+1
n=84
Find first term if common difference last term and number of term given
1. Find first term if common difference is 2 and and last term is 240 and number of term is 24.
Sol. an=240 n=24 d=2 ,a=?
an = a + (n -1)d
240 = a + (24-1)2
240 = a+ (23)2
240 = a + 46
a= 240 -46
a = 194
Applied questions on A.P by using an=a+(n-1)d formula
1.in an A.P if 3rd term is 9 and 9th term is 27 find its 27th term.
Sol. Here given,
a3=9. and a9=27 a27=?
for finding A.P first of all we should find common difference and first term.
For finding d
d = (ap-aq)/(p-q)
ap=a3=9. And aq = a9=27
d= (9-27)/(3-9)
d=(-18)/(-6)
d=3-----------(1)
Now,
To find first term
a3 = 9
a + 2d = 9
a + 2(3) = 9
a + 6 =9
a =9-6
a = 3--------------(2)
Now
Finding 29th term
a29 = a + 28d
a29 = 3 + 28(3)
a29 = 3 + 84
a29 =87
2. Find 27 th term if first term is 12 and 9th term is 108.
Sol. a=12 and a9=108. a27=?
a9 = 108
a + 8d = 108
12 + 8d =108
8d = 108 - 12
8d = 96
d = 96/8
d = 12
a27 = a + 26d
= 12 + 26(12)
=12 +312
=324
3.find common difference A.P. if 7th term exceeds 12th term by 5.
Sol. Given a7=a12+5. And d=?
a+6d=a+11d+5
6d-11d=5
-5d=5
d=5/-5
d=-1
4. The sum of 2nd and 6th term of A.P is 24 and 7th term is 21 find A.P
Sol. Given a2 + a6 = 24 and a7 = 21
a+d+a+5d=24
2a+6d=24
2(a+3d)=24
a+3d=24/2
a+3d=12-------(1)
a7 = 21
a + 6d = 21
a=21-6d--------(2)
Substitute a in (1)
a +3d=12
21-6d+3d =12
-3d=12-21
-3d = -9
d = 9/3
d=3
Substitute d in (2)
a = 21-6d
a = 21 -6(3)
a = 21 - 18
a = 3
Therefore the A.p is
3,3+3,3+2(3)......
3,6,9.......
5. If first term of an A.P is 12 and 3rd term is 26 find its 2nd term.
Sol. Given a=12. And a3 = 26 a2=?
d = (ap-aq)/(p-q)
d =(12-26)/(1-3)
d= -14/-2
d=7
a2=a+d
a2=12+7
a2=19
5. If two A.Ps have same common difference ,then difference of their 100th term is 100,then find difference between their 1000th term.
Sol. Let 1st A.P be
a1,a2,a3.......
2nd A.p be
b1,b2,b3......
According to the question
a100 -b100 = 100
a + 99d - (b + 99d) = 100
a + 99d - b -99d =100
a -b = 100---------------(1)
Now,
a1000 - b1000 = a + 999d - (b + 999d)
= a + 999d - b - 999d
=a - b
= 100. (By equation 2)
6. Find number of terms multiple 4 between 10 to 250
Sol. Terms multiple of 4 between 10 to 250 are
12,16,20..................248
a=12 d=a2-a1. an = 248. n= ?
d=16-12
d=4
an = a + (n-1)d
248= 12 + (n -1)4
248-12=(n-1)4
236=(n-1)4
(n-1)=236/4
n-1=59
n=59+1
n=60
Therefore 60 numbers multiple of 4 are there between 10 to 250
7.find number of 3 digit numbers divisible by 7.
Sol. 3 digit numbers divisible by 7 are 105,112,119..................994
a=105 d = a2 - a1. an = 994. n = ?
d = 112 - 105
d = 7
an = a + ( n-1)d
994 = 105 + (n-1)7
994-105=(n-1)7
889=(n-1)7
(n-1)=889/7
n-1=127
n = 127+1
n =128
There are 128 ,3digit numbers divisible by 7
Like this we can find first term ,common difference and last term and number of terms.
For video explanation visit my channel by clicking below link
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