There can be series where all the terms are related to the squares of numbers or cubes of numbers. With squares/cubes of numbers as the basis, there can be many variations in the pattern of the series. Let us look at various possibilities of series based on Squares /cubes. Each term of the series may be the square of a natural .

Number, such as 1, 4, 9,16,………………

1, 4, 9, 16

1^{2 }2^{2} 3^{2} 4^{2}

The numbers are squares of 1, 2, 3, 4 .... Respectively. The number which follows 16 (which is the square of 4) will be 25 (which is the square of 5). The terms of the series may be the squares of odd numbers (for example, 1, 9, 25, 49, ... . ..) or even numbers (for example, 4, 16, 36, 64, ... . ...). The terms of the series could be such that a number and its square are both given one after the other and such pairs are given in some specific pattern. For example, take the series 2, 4, 3, 9, 4, 16, .

2, 4, 3, 9, 4, 16

Here, 2 is followed by its square 4; then comes the number 3 (which is one more than 2) followed by its square 9 and so on. Hence, the next number in the series is 5 and the one after that is its square i.e., 25. Similarly, each term could be the square root of its predecessor. For example, in the series 81, 9, 64, 8, 49, 7, 36, ...., 81 is the square of 9, 64 the square of 8, and so on. Therefore the next number which follows in the series should be the square root of 36, i.e., 6. The terms of the series could be the squares of natural numbers increased or reduced by certain number. For example, in the series 3, 8, 15, 24, .

3 , 8, 15, 24

¯ ¯ ¯ ¯

2^{2}-1 3^{2}-1 4^{2}-1 5^{2}-1

We have {Squares of natural numbers - 1) as the terms. The first term is 22- 1; the second term is 32- 1; the third term is 42 - 1 and so on. Hence, the next term will be 62 - 1, i.e., 35 [Please note that the above series can also be looked at as a series with increasing differences. The differences between the 1" & 2' terms, the 2nd 7, 9, and & 3'd terms, and so on are 5, so on. Hence, the next difference should be 11 giving us the next term as 35]. There could also be a series with {squares of natural numbers + some constant). Like we have seen series with squares of numbers, we can have similar series with cubes of numbers. For example, take the series 1, 8, 27, 64, .... .

1, 8, 27, 64

¯ ¯ ¯ ¯

1^{3} 2^{3} 3^{3} 4^{3}

Here, all the terms
are cubes of natural numbers. So, the next term will be 5^{3}, i.e.,
125; .

Consider the series 2, 9, 28, 65, ..... .

2 , 9, 28, 65

¯ ¯ ¯ ¯

1^{3}+1 2^{3}+1 3^{3}+1 4^{3}+1

Here, the terms are {Cubes of natural numbers + 1}. The first term is 11+ 1; the second term is 2' + 1; the third term is 3' + 1 and so on. Hence the next term will be 53+ 1, i.e., 126.