Find the sum of the first $75$ terms of the arithmetic sequence that starts $5, 8, 11, \ldots$

Question

Find the sum of the first $75$ terms of the arithmetic sequence that starts $5, 8, 11, \ldots$

The answer is $8700$.

I found a formula to be $3x+2$. So the $1$st term is $$3(1)+2=5$$ 2nd term $$3(2)+2=8$$ 3rd term $$3(3)+2=11$$ And so on to the 75th term $$3(75)+2=227$$ I did not get the right answer? What did I do wrong? Please help?

Answer 1

$$u_1=5$$ $$u_2=8=5+3$$ $$u_3=11=5+2.3$$ $$u_{75}=5+74.3=227$$

$$S=5+8+11+...+224+227$$ $$S=227+224+...8+5$$ by sum $$2S=232+232+...232=232.75$$

$$S=116.75=8700$$

Answer 2

The question didn't ask for the 75th term, it asked for the sum of the first 75 terms. So instead of just plugging $75$ into $$3x+2$$ You instead need to calculate $$\sum_{x=1}^{75} 3x+2$$

Also, the formula $$\sum_{x=1}^n x=\frac{n(n+1)}{2}$$ may be helpful to you here.