Question

If the sum to \[2n\] terms of an AP \[2,4,6,8,...\] is equal to the sum of \[n\] terms of an AP \[57,59,61,63,...\] then \[n\] is equal to
A. \[10\]
B. \[18\]
C. \[12\]
D. \[13\]

Answer 1

Hint: Arithmetic Progression (AP) is a sequence of numbers in order in which the difference of any two consecutive numbers is a constant value.Use the formula of sum to \[2n\] terms of an AP and sum of \[n\] terms of an AP as required in the question. Equate them and find the value for \[n\] . We use the formula for n terms if an AP is \[{S_n} = \dfrac{n}{2}\left( {2a + (n - 1)d} \right)\] .

Complete step-by-step answer:
We can define arithmetic progression in many ways. A mathematical sequence in which the difference between two consecutive terms is always a constant and it is abbreviated as AP.
An AP is defined as a sequence of numbers in which for every pair of consecutive terms, the second number is obtained by adding a fixed number to the first one.
The fixed number that must be added to any term of an AP to get the next term is known as the common difference of the AP.
The sum of \[n\] terms of an arithmetic progression, \[{S_n} = \dfrac{n}{2}\left( {2a + (n - 1)d} \right)\]
Now sum of \[2n\] terms of an AP \[2,4,6,8,,...\] is given by :
Sum \[ = \dfrac{{2n}}{2}\left( {2 \times 2 + (2n - 1)2} \right)\]
 \[ = n(2 + 4n)\]
Sum of \[n\] terms of an AP \[57,59,61,63,...\]
Sum \[ = \dfrac{n}{2}\left( {2 \times 57 + (n - 1)2} \right)\]
 \[ = n(56 + n)\]
We are given that the sum to \[2n\] terms of an AP \[2,5,8,11,...\] is equal to the sum of \[n\] terms of an AP \[57,59,61,63,...\]
Therefore we get \[n(1 + 6n) = n(56 + n)\]
 \[2n + 4{n^2} = 56n + {n^2}\]
On solving the like terms together we get
 \[3{n^2} = 54n\]
On solving the equation we get
 \[n = 18\]
So, the correct answer is “Option B”.

Note: Use the formula of sum to \[2n\] terms of an AP and sum of \[n\] terms of an AP as required in the question very carefully. Equate them accordingly to find the value for \[n\] .Keep in mind that value for \[n\] should only be a positive whole number. Take care of the calculations in the problem.