Question

Observe the following pattern
 $
  {2^2} - {1^2} = 2 + 1 \\
  {3^2} - {2^2} = 3 + 2 \\
  {4^2} - {3^2} = 4 + 3 \\
  {5^2} - {4^2} = 5 + 4 \\
  $
And find the value of $ {100^2} - {99^2} $ .

Answer 1

Hint: Here, we are given a pattern and based on that pattern we need to find the value for $ {100^2} - {99^2} $ . Now, this is a logical question, so to find the value of $ {100^2} - {99^2} $ , we need to find the logic behind this pattern. The logic behind this pattern is that the difference of the squares of two consecutive numbers is equal to the sum of these numbers. Using this logic, we are going to find the value of $ {100^2} - {99^2} $ .

Complete step-by-step answer:
In this question, we are given a pattern and based on that pattern we need to find the value of $ {100^2} - {99^2} $ .
The given pattern is:
 $
  {2^2} - {1^2} = 2 + 1 \\
  {3^2} - {2^2} = 3 + 2 \\
  {4^2} - {3^2} = 4 + 3 \\
  {5^2} - {4^2} = 5 + 4 \;
  $
Now, this is a logical question, and to find the value for $ {100^2} - {99^2} $ , we need to understand the logic behind the given pattern and then use that logic to find the required answer.
Let us find the logic behind the given pattern.
Here, we can see that the square of the number being subtracted is written on right is smaller than the square of number written on right and the result is given by their addition.
That means the logic behind this pattern is that when we subtract the squares of two consecutive numbers, where the smaller number is subtracted from the bigger number, the answer will be the sum of these consecutive numbers.
 $ {2^2} - {1^2} = 2 + 1 $ : The difference of squares of two consecutive numbers $ \left( {{2^2} - {1^2}} \right) $ is equal to the sum of these consecutive numbers $ \left( {2 + 1} \right) $ .
 $ {3^2} - {2^2} = 3 + 2 $ : The difference of squares of two consecutive numbers $ \left( {{3^2} - {2^2}} \right) $ is equal to the sum of these consecutive numbers $ \left( {3 + 2} \right) $ .
 $ {4^2} - {3^2} = 4 + 3 $ : The difference of squares of two consecutive numbers $ \left( {{4^2} - {3^2}} \right) $ is equal to the sum of these consecutive numbers $ \left( {4 + 3} \right) $ .
 $ {5^2} - {4^2} = 5 + 4 $ : The difference of squares of two consecutive numbers $ \left( {{5^2} - {4^2}} \right) $ is equal to the sum of these consecutive numbers $ \left( {5 + 4} \right) $ .
Hence, according to this logic, the answer of $ {100^2} - {99^2} $ will be equal to the sum of the numbers that is 100 and 99. Therefore,
 $ \Rightarrow {100^2} - {99^2} = 100 + 99 = 199 $
So, the correct answer is “199”.

Note: Let us find these answers using the normal method and see if we are correct or not.
 $
  {2^2} - {1^2} = 4 - 1 = 3 = 2 + 1 \\
  {3^2} - {2^2} = 9 - 4 = 5 = 3 + 2 \\
  {4^2} - {3^2} = 16 - 9 = 7 = 4 + 3 \\
  {5^2} - {4^2} = 25 - 16 = 9 = 5 + 4 \;
  $
Therefore,
 $ \Rightarrow {100^2} - {99^2} = 10000 - 9801 = 199 = 100 + 99 $
Hence, our answer is correct.