Filling the Next Perfect Square in the Sequence

Mathematics is a beautiful language that reveals itself in patterns and sequences. One such fascinating sequence is the perfect squares, which are obtained by squaring the positive integers. This article will guide you through understanding the pattern, solving 1^2, 4, 9, 25, 36, and determining the next number in this sequence.

Understanding the Perfect Squares Sequence

The given sequence:

1^2 1 2^2 4 3^2 9 4^2 16 5^2 25 6^2 36

Each term in this sequence represents the square of consecutive integers. To find the next number in the sequence, we simply square the next integer after 6.

Calculating the Next Term

The next integer after 6 is 7. Squaring 7 gives us:

7^2 49

Therefore, the next number in the sequence is:

7^2 49

Alternative Methods to Verify the Sequence

Let's explore a few alternative methods:

Subtraction Method

Email: 1 4 34 9 59 16 716 25 925 36 1136 13 64

Explanation:

Let's break it down:

1 - 1 0, but the given sequence is 3 4 - 1 3 9 - 4 5 16 - 9 7 25 - 16 9 36 - 25 11 36 13 49 (This is correct)

General Formula Method

The nth term of the sequence is given by:

an n2

For the sequence 1, 4, 9, 16, 25, 36, the next term is:

72 49

The Sequence: 1, 4, 9, 16, 25, 36, 49, ...

This sequence is the first six perfect squares, and the next perfect square, when continuing the pattern, is 49. This sequence of perfect squares can be generalized as:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, ...

Each term is the square of an integer, starting from 1.

Conclusion

The next number in the sequence 1, 4, 9, 25, 36 is clearly 49. This is both a straightforward calculation and a fascinating exploration of the beauty of mathematics in sequences and patterns.