Question

Make a conjecture about the $n$th term of the following series.

• A. 2
• B. 2${}^{n-1}$
• C. 3n - 2

Answer 1

Let us make a conjecture for the $n$th terms of the following series.

2, 2, 2, 2, 2 ....
Here, it can be clearly observed that all of the terms are 2.
Therefore, the $n$th term of this series should also be 2.

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1, 2, 4, 8, 16 ....

1st term = 1 = 2${}^0$ = 2${}^{1-1}$
2nd term = 2 = 2${}^1$ = 2${}^{2-1}$
3rd term = 4 = 2${}^2$ = 2${}^{3-1}$
4th term = 8 = 2${}^3$ = 2${}^{4-1}$
5th term = 16 = 2${}^4$ = 2${}^{5-1}$

$n$th term = 2${}^{n-1}$

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1, 4, 7, 10, 13 ....

1st term = 1 = 3$\times$1 - 2
2nd term = 4 = 3$\times$2 - 2
3rd term = 7 = 3$\times$3 - 2
​​​​​​​4th term = 10 = 3$\times$4 - 2​​​​​​​​​​​​​
5th term = 13 = 3$\times$5 - 2

$n$th term = 3$n$ - 2