Explanation:
The formula:
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=
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2
(
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1
+
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)
S
n
=
2
n
(a
1
+a
n
)
is used to solve for the sum of the arithmetic sequence given the first term a₁, the number of terms n, and the last term in an.
Example:
3, 6, 9, 12, 15,...,123
The first term, a₁ = 3
The last term an = 123
Common difference, d = 3 (because the sequence are multiples of 3)
Number of terms, n= ?
Find the number of terms, n:
an = a₁ + (n-1) (d)
123 = 3 + (n-1) (3)
123 = 3 - 3 + 3n
123/3 = 3n/3
n = 41
To find the sum of the given sequence without adding 3 + 6 + 9, ... + 123, we use the formula:
S₄₁ = (41/2) (3 + 123)
S₄₁ = (41/2) (126)
S₄₁ = (41)(63)
S₄₁ = 2,583 ⇒ the sum of the given sequence