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Arithmetic Progression: Sum of a Finite Number
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Arithmetic Progression: Sum of a Finite Number
In Arithmetic Progression, every successive number in the list is formed by adding the same fixed number to the previous number (The fixed number may be any real number). This fixed number is known as common difference of an AP.
Sum of the finite number of terms of an AP is given by:

where,
a = First term of AP
n = Total number of terms
l = last term of AP given by l = a + (n-1)d
d = common difference
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Example: The common difference of an AP is -1. Find its sum, if its first term is 100 and the last term is -10.
Solution: In the usual notation a1 = 100.
let the number of terms be n. Then an = -10.
Here value of n is to be found out. Now
an = a­1 + (n - 1)d
-10 = 100 + (n - l)(-1)
or (n - l)= 100 + 10
or n = 111
Thus, -10 is the 111th term. Thus, we have to find the sum of the first 111 terms of an AP whose first term is 100, and 111th term is -10.

s111 = 111{(100 + (-10)) / 2}, Using
= 111 x 45 = 4995

Example: Find the sum of the first 24 terms of the sequence whose nth term is given by
an = 3 + (2/3)n
Solution: Let us examine whether the given sequence is an AP. It would be an AP, provided the difference between any two successive terms is a fixed number. Now
an = 3 + (2/3)n
an + 1 = 3 (2/3)(n + 1)
= 3 + (2/3)n + (2/3)
= [3 + (2/3)n] + (2/3)
= an+ (2/3)
an + 1 - an = (2/3), a fixed number.

Hence, the given sequence is an AP with common difference = 2/3.
Thus, an = 3 + (2/3) = 11/3
Hence, the sequence is
11/3, 13/3, 15/3, 17/3, ....
The sum to 24 terms is given by
S24 = 24 {(11/3) + (1/2) x (24 - 1) x (2/3)}
= 24{(11/3) + (23/3)}
= 24 x (34/3) = 8 x 34 = 272
 
   
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