Answer :
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.
Calculate the sum of the Arithmetic Series:
- 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 = [tex]\frac{n}{2} (2a+(n-1)d)[/tex]
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 = [tex]\frac{n}{2} (a_{1}+a_{n} )[/tex]
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 = [tex]\frac{n}{2} (2a+(n-1)d)[/tex]
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 = [tex]\frac{n}{2} (a_{1}+a_{n} )[/tex]
Sₙ = 8/2(1765 - 692)
Sₙ = 4(1073)
Sₙ = 4292
Hence sum of the Arithmetic Series 1765,1414,...,-692 is 4292.
Learn more about Arithmetic Series here:
brainly.com/question/10396151
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