The Sum of 40 terms of the Arithmetic Sequence is 3,420..
Arithmetic Series is defined as a sum of the terms of arithmetic progression or sequence.
An Arithmetic progression is a sequence of terms in which the difference between two consecutive terms is a constant number.
We have following information about series
sum of 12 terms of an arithmetic series (S₁₂) = 186
the 20th term of arithmetic sequence = 83
i.e., a₂₀ = 83
As we know that nth term of arithmetic progression is aₙ= a + (n-1)d
where , a---> first term of sequence
n ----> nth term
d -----> constant difference between two consecutive terms
put n= 20 in above formula we get,
83 = a + (20-1 )d = a+ 19d ----(1)
Now , Using the sum of n arithmetic terms formula, Sₙ = n/2 ( 2a + (n-1)d)
so, for n = 12
186 = 12/2 ( 2a + 11d ) => 2a + 11d = 31 ---(2)
Solving equations (1) and (2) we get,
27d = 135 => d = 5
putting the value of d in equation (1)
83 = a + 19×5 => a = 83 - 95 = -12
thus sequence is like as -12, -7 , -2 , ------
Sum of first 40 terms of arithmetic sequence (S₄₀) = 40/2 ( 2× (-12) + 39d ) = 20 ( -24 + 39×5)
= 20( 171) = 3,420
So, required result is 3,420..
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