# Incorrect partial sum formula in textbook?

I was helping my brother with his maths homework, where he has just started learning about arithmetic series and their formulas such as the sum of the first $$n$$ terms ($$S_n$$) or finding the $$n$$th term of the sequence ($$T_n$$). While looking through some of the questions in his textbook, I came across this question:

"Find the original sequence $$T_n$$ if its partial sums $$S_n$$ are given by $$S_n = n^2+5$$"

We can find $$T_n$$ if we take $$S_n-S_{n-1}$$ :

$$S_n-S_{n-1}=n^2+5 - (n-1)^2-5=2n-1$$

$$\Rightarrow T_n =2n-1$$ which is the expected solution to the question by the textbook. However upon trying to find an actual sequence of numbers that fit the solution I came up empty handed! I feel as if there is no way the sum can have a constant term but I'm not sure why.

Is the textbook's solution actually correct? Is there such a series of numbers that works?

If the textbook's solution is wrong, am I correct in assuming that for any arithmetic series of numbers, the sum of the series cannot be of the form $$S_n=f(n)+k$$, where $$k$$ is a constant?

• I guess you can define the $0^{th}$ term as 5? Commented Nov 18, 2023 at 7:22
• For partial sums, the usual convention is $S_0 = \sum_{i = 1}^{0}T_i = 0$. So $S_0 = 5$ is impossible. Commented Nov 18, 2023 at 21:21

$$T_n=S_n-S_{n-1}$$ is true only when $$n\ge2$$. It's not true when $$n=1$$, because then we would have $$T_1=S_1-S_0$$ but $$S_0$$ is not defined.
So for $$n\ge2$$ we have $$T_n=2n-1$$ as you showed. For $$T_1$$, we have $$T_1=S_1=1^2+5=6$$.
So the first few terms of the sequence are $$6,3,5,7,9,\dots$$. The sequence is not arithmetic. But if we ignore the first term, then it's arithmetic.
The sum of an arithmetic series is $$S_n=\frac{n}{2}(T_1+(n-1)d)$$ where $$T_1$$ is the first term, and $$d$$ is the common difference. This gives $$S_n=\frac{d}{2}n^2+\frac12(T_1-d)n$$. So if $$S_n=f(n)+k$$ (where $$f(n)$$ itself has no constant term) then $$k$$ must equal $$0$$.