Sequences and Series: arithmetic squence based

Question

I have this question on sequence and series that i don't think it should be difficult, but I am running into problems.

The question is this:

The first 10 terms of an arithmetic sequence adds to 250. Find the explicit form t(n) for this arithmetic sequence. All terms t(n) are Natural Numbers.

I was able to get the solution but I had to basically guess the solution because I ended up with one equation with 2 unknowns.

I am hoping that there is a full algebraic way of solving this without having to guess the numbers required to solve it.

Hope someone can help.

Answer 1

If you let the first term be $a$ and the difference be $d$ we have $$10a+45d=250\\2a+9d=50$$ Now $d$ must be even and less than $6$, leaving only $2$ and $4$. If you think $0$ is a natural, add that in. This gives two or three solutions $$d=4,a=7\\d=2,a=16\\d=0,a=25$$ Diophantine equations often have fewer equations than variables. You need to make use of the fact that the variables are naturals or integers to get the solution(s).

Answer 2

We have that

$$S_n=\frac{n(a_1+a_n)}2=5(a_1+a_n)=250 \implies a_1+a_n=50$$

and for $a_{i+1}=a_i+d$

$$d=\frac{a_n-a_1}{n-1}=\frac{a_n-a_1}{9}=\frac{50-2a_1}{9}$$

that is

$$50-2a_1\equiv 0 \mod 9 \implies 2a_1-5\equiv 0 \mod 9\implies2a_1=5+k9$$

which gives the solutions

• $a_1=7,16,25$