Arithmetic Sequence Equation

Question

Find a solution for: $$ \left( x+1 \right) + \left( x+4 \right) + \left( x+7 \right) +...+ \left( x+28 \right) = 155 $$

I tried with $S_n$ formula, $$S_n = 155 = \frac{n[(x+1)+(3n-2)]}{2}$$ But I suppose it's not right.

I tried also putting numbers in $x$ from $1$ and so on until the correct answer, and I got an answer, but I want to know an easier way when that random number will be higher than a digit.

Answer 1

The sum can be written as $$\sum_{k=0}^9(x+1+3k)$$ Now the sum of consecutive terms of an arithmetic sequence is the average of the first and the last term, times the number of terms. Namely, in the present case: $$\frac{(x+1)+(x+28)}2\times 10=(2x+29)\times 5=10x+145,$$ so we obtain $$10x+145=155\iff 10x=10\iff x=1.$$

Answer 2

$$(x+1)+(x+4)+(x+7)+(x+10)+(x+13)+(x+16)+(x+19)+(x+22)+(x+25)+(x+28)=155$$ $\implies$ $$10x+145=155$$ $\implies$ $$10x=10$$ $\implies $ $$x=1$$