Series

[mathmajor]

Is there a shortcut of doing "The 10th term of 6,9,14,21,30,..."

[RussiaPSJA]

The sequence is x^2+5, where x is the xth term.

So, the 1st term is 1^2 + 5 = 6
2nd term is 2^2 + 5 = 9
3rd term is 3^2 + 5 = 14...

Thus the 10th term is 10^2 + 5, which is 105.

[mathmajor]

Thanks! I always saw that it added 3, then 5, then 7, and so on. This way is much faster.

[RussiaPSJA]

There's that route, and then there's this route, although this one isn't probably something you'd want to do during a number sense test.

You noticed that the differences were 3, 5, 7... So they were going up by 2 each time.

So, the common difference is two.

The last term in this sequence is going to be the number of spaces * 2 (the common difference) + the first term, and because these are already spaces that we're dealing with, we know there's 9 of them (9 spaces for the 10 original terms), and so these 9 spaces are going to have 8 spaces of their own.

Doing the calculation, 8 spaces * 2 + 3 = 16 + 3 = 19. 19 is the last term.

Now, we simply multiply the average of the first and last term by the number of terms to obtain what we need from this, but then we need to add the first term of the original sequence so that we could obtain our answer.

[The average] * [The number of terms] + [The first term in the original sequence] = (3+19)/2 * 9 + 6 = 11 * 9 + 6 = 99 + 6 = 105, your answer.

That's another way of working it without realizing that the pattern is x^2 + 5.