Around the World in 80 Questions (Day 9): The sum of the first n term

Question

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

kratos0906 · Accepted Answer

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

Kinshook · Answer

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

Jarvis07 · Answer

Good question as always.
 S_n= 29n - n^2 
We can find the first term by putting n=1
 a_1=29*1-1^2=28 
 S_2=29*2-2^2=54 . Hence,  a_2=54-28=26 
Hence, commond difference=26-28=-2
Therefore, the AP is: 28, 26, 24... 
to find the term of 4:
 4=28+(n-1)*(-2) 
 n=13. 
Hence B is correct.

missionmba2025 · Answer

S1 = First Term = T1

S1 = 29 - 1 = T1 = 28

S2 = 29*2 - 4 = 58 - 4 = 54

T2 = S2 - S1 = 26

Difference = d = -2

an = T1 + (n-1)d

4 = 28 + (n - 1) * (-2)

-24 = (n - 1)* (-2)

12 = n - 1

n = 12+1 = 13

IMO B

BottomJee · Answer

S_1  =  a_1  = (29*1) -  1^2  = 29 - 1 = 28
 S_2  =  a_1  +  a_2  = (29*2) -  2^2  = 58 - 4 = 54
Therefore,  S_2 = S_1 + a_2 
=>  54 = 28 + a_2 
=>  54 - 28 = a_2 
=>  26 = a_2 
Since, the sequence is A.P.
then difference between two terms,d = a_2 - a_1 = 26 - 28 = -2

Substituting the values of  a_n  =  a_k  = 4, d and  a_1  in the formula  a_n = a_1 + (n-1)d , we get
4 = 28 + (k-1)*(-2)
=> -24 = -2k + 2
=> 26 = 2k
=> 13 = k

Hence, OA should be B.

sujoykrdatta · Answer

Sum of the first n terms of an AP = (n/2)  where a is the first term and d is the common difference
Thus: (n/2)  = n(29 - n)
=> 2a + (n-1)d = 58 - 2n
=> (2a - d) + nd = 58 - 2n
Comparing: d = -2; 2a - d = 58 => a = (58 + d)/2 = 28

Thus, if the k-th term is 4: 28 + (k - 1)(-2) = 4
=> 28 - 2(k - 1) = 4
=> k - 1 = 12
=> k = 13

Answer B

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Around the World in 80 Questions (Day 9): The sum of the first n term

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