# Finding the formula for the sequence: 7, 10, 16, 25, etc, in terms of $n$, given…

In terms of $$n$$, find the formula for the sequence $$a_n$$: 7, 10, 16, 25; given that the difference between each adjacent element in the sequence creates an arithmetic sequence of its own.

I got the incorrect solution and I cannot trace my mistake:

We create a new sequence $$b_n$$ based on the given information:

$$b_1 = 10 - 7 = 3$$
$$b_2 = 16 - 10 = 6$$
$$b_3 = 25-16 = 9$$
$$d_{b_n} = 3$$

We know that: $$a_n - a_{n-1} = b_{n-1}$$ If we were to add all possible elements: $$A_n - A_{n-1} = B_{n-1}$$ We also know that $$A_n - A_{n-1} = a_n$$, so therefore $$a_n = B_{n-1}$$: $$a_n = \frac{[2b_1 + d(n-2)](n-1)}{2} = \boxed{\frac{3n(n-1)}{2}}$$ However, this is incorrect as it gives values inconsistent with the definition of the sequence. The correct answer is $$\frac{3n^2-3n+14}{2}$$.

Where have I gone wrong?

• I don't see where you used the initial condition $a_1=7$. Just knowing the differences between terms in a sequence does not determine the sequence uniquely. You still need some more information, like a starting value. – lulu Jan 11 '19 at 17:05
• @lulu Isn't it implied from $b_1 = 10 - 7 = 3$? – daedsidog Jan 11 '19 at 17:07
• No. Just knowing that $a_2-a_1=3$ doesn't tell you what $a_1,a_2$ are. – lulu Jan 11 '19 at 17:17

You have an error when adding the sequences up to $$n$$.
$$a_2 - a_1 = b_1 \\ a_3 - a_2 = b_2 \\ \dots \\ a_n - a_{n - 1} = b_{n - 1}$$
$$\sum_2^n a_i + \sum_1^{n-1}a_i = \sum_1^{n-1}b_i$$
The second term here is indeed $$A_{n-1}$$, and the right hand side is $$B_{n=1}$$, but the first is $$A_n - a_1$$. Therefore you have $$A_n - a_1 - A_{n-1} = B_{n-1}$$. Now, using $$A_n - A_{n-1} = a_n$$, we have $$a_n = B_{n-1} + a_1 = \frac{3n(n-1)}{2} + 7$$