Question

Solve the following expression : \[{11^3} + {12^3} + ... + {20^3}\]
A. Cannot be determined
B. is divisible by $5$ having negative sign
C. is divisible by \[10\]
D. is divisible by $5$

Answer 1

Hint: In the given problem, we need to find the sum of the given sequential series which involves some certain algebraic parameters such as cubes. So, to solve the desired expression, we will consider the formula for summation of the cubes of the first ‘n’ natural numbers ‘N’ (by considering the previous series of the given sequence) respectively.

Complete step by step answer:
The given expression implies sequential series,
\[{11^3} + {12^3} + ... + {20^3}\]
As a result, the series is the series containing the addition having cube of each term,
Since, we know that the sum of such cases are,
${S_n} = \dfrac{{n{{\left( {n + 1} \right)}^2}}}{2}$
where, ‘${S_n}$’ is the sum of the respective series, and ‘$n$’ is the last term in the respective series.

Now, considering the expression \[{11^3} + {12^3} + ... + {20^3}\]. But, the expression or sequence or series, is incomplete, hence, considering the similar previous series that is \[{1^3} + {2^3} + ... + {10^3}\], we get
\[{S_n} = \left( {{{11}^3} + {{12}^3} + ... + {{20}^3}} \right) - \left( {{1^3} + {2^3} + ... + {{10}^3}} \right)\]
Now, using the formula mentioned above
${S_n} = \dfrac{{n{{\left( {n + 1} \right)}^2}}}{2}$
We get
\[{S_n} = \dfrac{{20{{\left( {20 + 1} \right)}^2}}}{2} - \dfrac{{10{{\left( {10 + 1} \right)}^2}}}{2}\]

Solving the equation mathematically, we get
\[{S_n} = \dfrac{{20{{\left( {21} \right)}^2}}}{2} - \dfrac{{10{{\left( {11} \right)}^2}}}{2} \\
\Rightarrow {S_n} = \dfrac{{20\left( {441} \right)}}{2} - \dfrac{{10\left( {121} \right)}}{2} \\ \]
Hence, the required sum of the series is,
\[{S_n} = \dfrac{{8820}}{2} - \dfrac{{1210}}{2} \\
\Rightarrow {S_n} = \dfrac{{7610}}{2} \\
\therefore {S_n} = 3805 \]
As a result, it seems that the sum of the given series is divisible by $5$ that is ${S_n} = \dfrac{{3805}}{5} = 761$ where the remainder is ‘zero’!

Hence, option D is correct.

Note: For finding the certain sum of the sequential series problem algebraically, consider the previous term or a series\[{1^3} + {2^3} + ... + {10^3}\]for this problem particularly. As a result, to that term we have the generalized formula for the summation of cubes i.e.${S_n} = \dfrac{{n{{\left( {n + 1} \right)}^2}}}{2}$respectively. Learning these equations or formulae will be really helpful to solve these drastic problems as per sequential series is concerned. Care should be taken while solving the certain equations.