Question

If \[{S_n} = {1^3} + {2^3} + .... + {n^3}\] and \[{T_n} = 1 + 2 + 3 + ... + n\], then
A.\[{S_n} = {T_n}\]
B.\[{S_n} = {T^4}_n\]
C.\[{S_n} = {T^2}_n\]
D.\[{S_n} = {T^3}_n\]

Answer 1

Hint: This question is related to sequence and series , where \[{S_n} = {1^3} + {2^3} + .... + {n^3}\] and \[{T_n} = 1 + 2 + 3 + ... + n\] 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 : \[{S_n} = {1^3} + {2^3} + .... + {n^3}\] and \[{T_n} = 1 + 2 + 3 + ... + n\] .
The general formula for the series of cube of \[n\] natural number is given by \[ = {\left[ {\dfrac{{n\left( {n + 1} \right)}}{2}} \right]^2}\]. Therefore , the series \[{S_n}\] will be equals to \[{S_n} = {\left[ {\dfrac{{n\left( {n + 1} \right)}}{2}} \right]^2}\] …..equation (A) .
Similarly for the series \[{T_n} = 1 + 2 + 3 + ... + n\] , the general formula for sum of \[n\] natural numbers is given by \[ = \dfrac{{n\left( {n + 1} \right)}}{2}\] .
Therefore , the series \[{T_n}\] will be equals to \[{T_n} = \dfrac{{n\left( {n + 1} \right)}}{2}\] ……. Equation (B) .
The equation (A) can be written as \[{S_n} = {\left[ {{T_n}} \right]^2}\] , since \[{T_n} = \dfrac{{n\left( {n + 1} \right)}}{2}\] .
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 .