Question

Find the missing number using the given pattern.
$2^2+3^2+6^2=7^2{3}^2+4^2+{12}^2={13}^2{4}^2+5^2+{20}^2={21}^2$
What is the general form for the above pattern?

• A. $(m{)}^2+(m+1{)}^2+\left(\right(m)\times (m+1){)}^2=(\left(m\right)\times (m+1)+1{)}^2$
  
  
• B. $(m{)}^2+(m+1{)}^2+\left(\right(m)\times (m+1){)}^2=(\left(m\right)\times (m+1){)}^2$
  
• C. $(m{)}^2+(m+1{)}^2+\left(\right(m)\times (m){)}^2=(\left(m\right)\times (m+1)+1{)}^2$
  
• D. $(m{)}^2+(m{)}^2+\left(\right(m)\times (m+1){)}^2=(\left(m\right)\times (m+1)+1{)}^2$
  

Answer 1

The correct option is A $(m{)}^2+(m+1{)}^2+\left(\right(m)\times (m+1){)}^2=(\left(m\right)\times (m+1)+1{)}^2$



Base of first number is related to the base of second square numbers. They are consecutive.
Let the base of first number be m then,
The base of second number is m+1

We can see that the base of third number is related to the base of first and second square numbers.
Base of third number is the product of bases of first and second number.

$\therefore$ Base of third number is
$\left(m\right)\times (m+1)$

Base of resultant square number is related to the base of the third square number. The base of the resultant square number is the number next to the base of the third number.
$\therefore$ The base of the resultant square number is
the number next to $\left(m\right)\times (m+1)$ i.e. $\left(m\right)\times (m+1)+1$

$\therefore$ The general form of the sequence
$(m{)}^2+(m+1{)}^2+\left(\right(m)\times (m+1){)}^2=(\left(m\right)\times (m+1)+1{)}^2$