# 02 Basics of Arithmetic & Geometric Progression (Grades 9 to 12) Part-02

by Anil Satpute Mathematician, Writer, Poet
Dear Students,

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In the Previous Blog we had seen some important concepts of Arithmetic & Geometric Progression. Click HERE to Revise that Blog.

In this Blog, we Describe the Important Formulas of an AP & GP.
1) tn =  a + (n - 1) d
2) Sn = n * (a + l)/2
3) Sn = n * [ 2 a + (n - 1) d ) ] / 2

Now we will see some examples:

Problems related to tn =  a + (n - 1) d:

A) Find nth term of an AP 3, 5, 7, 9 ...

Solution:
1)  Here a = 3, d = (5 - 3) = 2 so, d = 2.
2)  We know that
tn =  a + (n -1) d
=  3 + (n -1) (2)
=  3 + (2 n - 2)
=  1 + (2 n)
=  (2 n) + 1
3) Answer: Here the nth term of an AP is  tn =  (2 n) + 1

B) Find first term of an AP in which d = 4 and it's 100th term is 403.

Solution:
1)  Here d = 4 and t100 =  403.
2)  We know that
tn =  a + (n -1) d
403 =  a + (100 - 1)*(4)
403 =  a + (99)*(4)
403 =  a + (396)
a =  403 - 396
a =  7
3) Answer: Here the 1st term is a = 7

C) If  nth term of an AP is m and mth term of an AP is n, then find the value of d.

Solution:
1)  Let " a " be the 1st term and " d " be the common difference.
2)  We know that
tn =  a + (n -1) d
3)  So we,
t =  a +  (n -1) d = m      ----------- (1)
tm =  a + (m -1) d = n       ----------- (2)
Subtract equation (2) from (1) we get,
a +  (n -1) d = m
a + (m -1) d =  n
(-)  (-)               (-)
----------------------------------
(n - 1 - m + 1) * d = (m - n)
(n - m) * d = (m - n)
d = (m - n)/(n-m)
d = - 1
4) Answer: Here common difference is d = - 1

For More details Please Click HERE.
Anil Baburao Satpute

### About Anil Satpute Mathematician, Writer, Poet

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