Given That 20=20+18+16 + . . . +X, How Many Terms Are On The Right Side Of The Equation?, Find The Sum Of 2212100201399201398201397 2013 . . . +100+10
Given that 20=20+18+16 + . . . +x, how many terms are on the right side of the equation?
Find the sum of −100–99–98–97 – . . . +100+101+102
Please answer these 2 questions.
Answer:
Step-by-step explanation:
Sum of arithmetic sequence:
Sn = n/2 2a₁ + (n-1)(d)
Where:
Sn = sum of the sequence
n = number of terms
a₁ = first term
d = common difference
Given:
20 = 20+18+16+...+x
Sn = 20
a₁ = 20
d = -2 (the sequence decreases by 2 per term)
x = nth term
Find n, the number of terms:
20 = n/2 2(20) + (n-1)(-2)
20 = n/2 {40 + 2 - 2n
20 = n 21 - n)
20 = -n² + 21n
Arrange the quadratic equation in standard form:
n² - 21n + 20 = 0
Factor:
(n - 20) (n - 1) = 0
n - 20 = 20
n = 20
n - 1 = 0
n = 1
Choose n = 20.
ANSWER: There are 20 terms.
Find the 20th term:
a₂₀ = 20 + (20-1)(-2)
a₂₀ = 20 + (19)(-2)
a₂₀ = 20 - 38
a₂₀ = -18
The 20th term x = -18.
Check:
Sn = n/2 (a₁ + an)
20 = 20/2 (20 + (-18))
20 = 10 (2)
20 = 20 (True)
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Find the sum of −100–99–98–97 – . . . +100+101+102
Where:
a₁ = -100
d = 1 (the sequence is increasing hence the common difference is positive)
an = 102
n = unknowm
Find n, the number of terms:
an = a₁ + (n-1)(d)
102 = -100 + (n-1)(1)
102 = -100 - 1 + n
102 = -101 + n
n = 102 + 101
n = 203
There are 203 terms.
Find the sum:
S₂₀₃ = 203/2 -100 + 102}
S₂₀₃ = 203/2 (2)
S₂₀₃ = 203
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