Hello how can you describe such a set: M = { 1 2 5 10 ...}
The gradient is always odd in steps of 2
Gradient 1 + 3 + 5 + 7 + ...
How can you describe such a set? I need somehow 2 variables to describe it so that x starts with 1 and goes to inf and g for the gradient which indicates that it have to be odd.
Theophiles answer is excellent but I thought I'd add this to help with similar cases:
The second difference is constant suggesting a quadratic nth term rule i.e. $T_n=an^2+bn+c$ \begin{array}{ccccccccc}1&&2&&5&&10... \\&1&&3&&5... \\&&2&&2...\end{array}
The coefficient of $n^2$ is twice the second difference or in this example $2a=2\implies a=1$
If the sequence was generated by $T_n=n^2$ alone the sequence would be $1,4,9,16...$ clearly something is missing. To compensate we see how this sequence differs from the one we want: \begin{array} &1&4&9&16... \\ 1&2&5&10... \\\hline 0&2&4&6... \\ \hline \end{array}
The sequence of the difference is $R_n=2n-2$
The sequence you want is therefore $T_n=n^2-(2n-2)$ or $$T_n=n^2-2n+2$$