Question
Statement - 1: The sum of the series $$1 + \left( {1 + 2 + 4} \right) + \left( {4 + 6 + 9} \right) + \left( {9 + 12 + 16} \right) + .....\left( {361 + 380 + 400} \right){\text{ is }}8000.$$
Statement - 2: $$\sum\limits_{k = 1}^n {\left( {{k^3} - {{\left( {k - 1} \right)}^3}} \right) = {n^3},} $$ for any natural number $$n$$ .
A.
Statement - 1 is false, Statement - 2 is true.
B.
Statement - 1 is true, Statement - 2 is true; Statement - 2 is a correct explanation for Statement - 1.
C.
Statement - 1 is true, Statement - 2 is true; Statement - 2 is not a correct explanation for Statement - 1.
D.
Statement - 1 is true, Statement - 2 is false.
Answer :
Statement - 1 is true, Statement - 2 is true; Statement - 2 is a correct explanation for Statement - 1.
Solution :
$${{n^{th}}}$$ term of the given series
$$\eqalign{
& = \,\,{T_n} = {\left( {n - 1} \right)^2} + \left( {n - 1} \right)n + {n^2} \cr
& = \,\,\frac{{\left( {{{\left( {n - 1} \right)}^3} - {n^3}} \right)}}{{\left( {n - 1} \right) - n}} \cr
& = \,\,{n^3} - {\left( {n - 1} \right)^3} \cr
& \Rightarrow \,\,{S_n} = \sum\limits_{k = 1}^n {\left[ {{k^3} - {{\left( {k - 1} \right)}^3}} \right]} \cr
& \Rightarrow \,\,8000 = {n^3} \cr
& \Rightarrow \,\,n = 20{\text{ which is a natural number}}{\text{.}} \cr
& {\text{Now, put }}n = 1,2,3,.....,20 \cr
& {T_1} = {1^3} - {0^3} \cr
& {T_2} = {2^3} - {1^3} \cr
& \cdot \cr
& \cdot \cr
& \cdot \cr
& {T_{20}} = {20^3} - {19^3} \cr
& {\text{Now, }}{T_1} + {T_2} + - - - + {T_{20}} = {S_{20}} \cr
& \Rightarrow \,\,{S_{20}} = {20^3} - {0^3} = 8000 \cr
& {\text{Hence, both the given statements are true and}} \cr
& {\text{statement 2 supports statement 1}}{\text{.}} \cr} $$