What is the range of the given function? {(–2, 0), (–4, –3), (2, –9), (0, 5), (–5, 7)}
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Sum of the Arithmetic Series 1765,1414,...,-692 is 4292. This can be obtained by first finding the number of terms using the formula of nth term and then finding the sum using the sum formula.
<h3>Calculate the sum of the Arithmetic Series:</h3>- nth term of a arithmetic sequence is given by the formula,
aₙ = a + (n - 1)d
where n is the number of terms, aₙ is the last term of the sequence, a is the first term of the arithmetic sequence and d is the common difference of the arithmetic sequence.
- Sum of arithmetic sequence formula is given by,
Sₙ =
where n is the number of terms, Sₙ is the sum of n terms, a is the first term of the arithmetic sequence and d is the common difference of the arithmetic sequence.
- If the last term aₙ is given sum of arithmetic sequence formula is given by,
Sₙ =
where n is the number of terms, Sₙ is the sum of n terms, a₁ is the first term of the arithmetic sequence and aₙ is the last term of the sequence.
Here in the question it is given that,
- The Arithmetic Series 1765,1414,...,-692
- a₁ = 1765, a₂ = 1414, aₙ = -692
We have to find the sum of the Arithmetic Series.
- First we have to find the common difference.
d = aₙ - aₙ₋₁
d = a₂ - a₁
d = 1414 - 1765 ⇒ d = -351
- Then we have to find the number of terms(n).
By using the formula of nth term we get,
aₙ = a + (n - 1)d
-692 = 1765 + (n - 1)(-351)
-692 = 1765 -351n + 351
351n = 1765 + 351 + 692
351n = 2808
n = 8
The number of terms n = 8
- Finally by using the formula of sum of arithmetic series we get,
Sₙ =
Sₙ = 8/2 (2(1765) + (8 - 1)(-351))
Sₙ = 4(3530 - 2457)
Sₙ = 4(1073)
Sₙ = 4292
Since last term of the arithmetic series is given we can also use the second formula,
Sₙ =
Sₙ = 8/2(1765 - 692)
Sₙ = 4(1073)
Sₙ = 4292
Hence sum of the Arithmetic Series 1765,1414,...,-692 is 4292.
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