Question

the 99th term of 2,7,14,23,34,.........is what?

Answer 1

The first level of differences are 5,7,9,11 and so on
The second level of differences is 2,2,2,2… [there can’t be a third level of differences as all the second level of differences are the same]
Now, given that there are two levels of differences, we can infer that the equation that represents this sequence will be quadratic [say there were three levels of differences, then it would have been a cubic equation]
KEY:
U - term
n - term no. [variable]
a,b,c - constants
U[n] = a [math]n^2[/math] + bn +c
In a sequence with two levels of differences, you can equate the first number in each level with certain expressions. For example, 2a is the expression for the first number in the second level of differences [although, as every number is 2, it doesn’t really matter - however, it’s better if you keep the habit].
If 2a = 2, then a = 1.
Now, the expression for the first number of the first level of difference, 5, the expression will be 3a+b. So,
3a+b = 5
3(1) + b = 5
Therefore, b = 2
Now, to find c, you’ll use the first term of the sequence itself. So for 2, the expression will be a+b+c [for cubic it’s a+b+c+d and so on]. So,
a + b + c = 2
(1) + (2) + c = 2
Therefore, c = -1.
So the equation will be U[n] = [math]n^2[/math] + 2n - 1
Input n as “99” to find the 99th term:
U[99] = [math]99^2[/math] + 2 (99) -1
= 9801 + 198 - 1
= 9998
Thus, the 99th term of the sequence will be
9998!