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27 Jul 2023, 08:00
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B
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Difficulty:
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Question Stats:
66% (02:45) correct
34%
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wrong
based on 662
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The sum of the first n terms of an arithmetic progression, \(a_1, \ a_2, \ ...\), is given by \(S_n= 29n - n^2\). If \(a_k = 4\), what is the value of k ? (An arithmetic progression is a sequence of numbers such that the difference between the consecutive terms is constant)
A. 12
B. 13
C. 14
D. 15
E. 100
A. 12
B. 13
C. 14
D. 15
E. 100
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27 Jul 2023, 08:39
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My Answer: B
a1=S1=28
S2=a1+a2=54 so, a2=26
d=a2-a1=-2
Now, ak=4=a1+(n-1)d
=28+(k-1)*-2
k=13
a1=S1=28
S2=a1+a2=54 so, a2=26
d=a2-a1=-2
Now, ak=4=a1+(n-1)d
=28+(k-1)*-2
k=13
Given: The sum of the first n terms of an arithmetic progression, \(a_1, \ a_2, \ ...\), is given by \(S_n= 29n - n^2\).
Asked: If \(a_k = 4\), what is the value of k ? (An arithmetic progression is a sequence of numbers such that the difference between the consecutive terms is constant)
\(S_n = 29n - n^2 \)
\(S_{n-1} = 29(n-1) - (n-1)^2\)
nth term a_n is difference between S_n & S_{n-1} since only a_n is added to S_{n-1} to get S_n
\(a_n = S_n - S_{n-1} = 29n - n^2 - {29(n-1) - (n-1)^2} = 29 - (2n-1) = 30 - 2n\)
\(a_k = 30 - 2k = 4\)
26 = 2k
k = 13
IMO B
Asked: If \(a_k = 4\), what is the value of k ? (An arithmetic progression is a sequence of numbers such that the difference between the consecutive terms is constant)
\(S_n = 29n - n^2 \)
\(S_{n-1} = 29(n-1) - (n-1)^2\)
nth term a_n is difference between S_n & S_{n-1} since only a_n is added to S_{n-1} to get S_n
\(a_n = S_n - S_{n-1} = 29n - n^2 - {29(n-1) - (n-1)^2} = 29 - (2n-1) = 30 - 2n\)
\(a_k = 30 - 2k = 4\)
26 = 2k
k = 13
IMO B
