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Question: If \[{S_n} = {1^3} + {2^3} + .... + {n^3}\] and \[{T_n} = 1 + 2 + 3 + ... + n\], then A.\[{S_n} = ...

If Sn=13+23+....+n3{S_n} = {1^3} + {2^3} + .... + {n^3} and Tn=1+2+3+...+n{T_n} = 1 + 2 + 3 + ... + n, then
A.Sn=Tn{S_n} = {T_n}
B.Sn=T4n{S_n} = {T^4}_n
C.Sn=T2n{S_n} = {T^2}_n
D.Sn=T3n{S_n} = {T^3}_n

Explanation

Solution

Hint : This question is related to sequence and series , where Sn=13+23+....+n3{S_n} = {1^3} + {2^3} + .... + {n^3} and Tn=1+2+3+...+n{T_n} = 1 + 2 + 3 + ... + n represents two different series . To solve questions related to series there is a general formula corresponding to different series .

Complete step-by-step answer :
Given : Sn=13+23+....+n3{S_n} = {1^3} + {2^3} + .... + {n^3} and Tn=1+2+3+...+n{T_n} = 1 + 2 + 3 + ... + n .
The general formula for the series of cube of nn natural number is given by =[n(n+1)2]2 = {\left[ {\dfrac{{n\left( {n + 1} \right)}}{2}} \right]^2}. Therefore , the series Sn{S_n} will be equals to Sn=[n(n+1)2]2{S_n} = {\left[ {\dfrac{{n\left( {n + 1} \right)}}{2}} \right]^2} …..equation (A) .
Similarly for the series Tn=1+2+3+...+n{T_n} = 1 + 2 + 3 + ... + n , the general formula for sum of nn natural numbers is given by =n(n+1)2 = \dfrac{{n\left( {n + 1} \right)}}{2} .
Therefore , the series Tn{T_n} will be equals to Tn=n(n+1)2{T_n} = \dfrac{{n\left( {n + 1} \right)}}{2} ……. Equation (B) .
The equation (A) can be written as Sn=[Tn]2{S_n} = {\left[ {{T_n}} \right]^2} , since Tn=n(n+1)2{T_n} = \dfrac{{n\left( {n + 1} \right)}}{2} .
Therefore , the correct answer for this question is option (C) .
So, the correct answer is “Option C”.

Note : Sequence and Series is one of the important topics in Mathematics . Though many students get confused between them , these two can be easily differentiated . Sequence and series can be differentiated , in which the order of sequence always matters in the sequence but it’s not the case with series.
Sequence and series are the two important topics which deal with the enumeration of elements . It is used in the recognition of patterns , for example, identifying the pattern of prime numbers , solving puzzles, and so on. Also, the series plays an important role in the differential equations and in the process of analyzing .
Sequence - The sequence is defined as the list of elements which are arranged in a specific pattern .
Series - The series is defined as the sum of the sequence Irrespective of the order of the sequence .