Question

How do you find the four consecutive integers whose sum is twice the cube of $5$?

Answer 1

Hint: We are given the sum of four consecutive integers same as twice the cube of $5$, we will start our solution by first undertaking what are consecutive numbers then we will let that first number to be $x$ then we will find the other by the definition of consecutive number, we will now find the cube of $5$. The sum of these four is the same as $2$- times the cube of $5$. So, we compare and solve for $x$, once we get $x$ (highest number), we will easily find another number.

Complete step by step solution:
We are given that four consecutive number sum is $2$ times the cube of $5$, so we first learn about consecutive numbers are those numbers which occur one after another for example $2\,\operatorname{and}\,\,3$ or $19\,and\,\,20$. All many more.
In general, $x,x+1$, they are consecutive as it just increased by $1$.
Now let our smallest number out of four be $x$, then the next number will be $x+1$ the third term will be $(x+1)+1$ which is $x+2$.
The fourth term will be $((x+)+1)+1$ which is $x+3$
So, we get that first number $x$
Second number is $x+1$
Third number is $x+2$
Fourth number is $x+3$
Now cube of $5$ means multiplication of $5$, $3$ time with its coefficient.
$\begin{align}
  & \Rightarrow {{5}^{3}}=5\times 5\times 5 \\
 & =125 \\
\end{align}$
As we know sum of these $4$ number is twice the cube of $5$
$\Rightarrow x+(x+1)+(x+2)+(x+3)=2\times 125$
Opening brackets, we get,
$x+x+x+1+2+3=250$
Now adding like terms, we get
$4x+6=250$
Now we subtract $6$ on both sides,
$\Rightarrow 4x+6-6=250-6$
Here,
$4x=244$
Now dividing both side by $4$
$\dfrac{4x}{4}=\dfrac{244}{4}$
So, we get,
$\Rightarrow x=61$
So our first number is $61$
Second number will be $x+1$
$\Rightarrow 61+1=62$
Third number is $x+2$
$\Rightarrow 61+2=63$
Fourth number is $x+3$
$\Rightarrow 61+3=64$

Note: Remember that consecutive term develops by preceding term, so we just add to the term to get a series of consecutive terms, also remember that we can never gain unlike term, that is we cannot add $x+2=2x$ or $2x+2=4x$. This is not correct, only like terms can be added up, similarly when we subtract, we follow the same rule to simplify also we need to carefully handle algebraic tools.