Question

What is the next number in the sequence 225, 100, 36, 9, 1 ?
(a) -7
(b) -6
(c) 0
(d) -1

Answer 1

Hint: First realise that all of the elements of this sequence are perfect squares of integers. Next, find the difference between the roots of these elements to find a pattern in these differences. Use this pattern to find the root of the next element and square that root to find the final answer.

Complete step-by-step answer:
In this question, we are given the sequence 225, 100, 36, 9, 1, …
We need to find the next term in this sequence.
We can see from the sequence that all of the elements of this sequence are perfect squares of integers.
225 = 15 $\times $ 15
100 = 10 $\times $ 10
36 = 6 $\times $ 6
9 = 3 $\times $ 3
1 = 1 $\times $ 1
So, following this pattern, we know that the next number will also be a perfect square.
Now, let us see the numbers whose perfect squares these elements are.
Let us write a new sequence with the roots of the elements of the original sequence to find a pattern in them.
15, 10, 6, 3, 1, …
From this sequence, we can see that:
15 – 10 = 5
10 – 6 = 4
6 – 3 = 3
3 – 1 = 2
As we can see, the difference between the consecutive numbers reduces by one after each step.
So, the difference between 1 and the root of the next element will be 1.
So the root of the next element will be: 1 – 1 = 0
The square of 0 is 0.
So, following the pattern, the next element of the sequence 225, 100, 36, 9, 1, … is 0.
Hence, option (c) is correct.

Note: In this question, you can find the answer in a shorter time also by using the options given. After the first step, we concluded that all of the elements of this sequence are perfect squares of integers and so the next element should also be a perfect square. Now, perfect squares cannot be negative. So, all the options are eliminated except option (c) which is our final answer.