[Math] Calculating the number of terms in arithmetic sequence

sequences-and-series

I know that if I have a set of numbers, let's say+

$1,2,3,4,5$
I can find the number of terms by subtraction the last term $5$ from the first terms $1$ and then add $1$: $(5-1)+1 = 5$, then the number of terms = 5.This is pretty much easy example just to make my question clearer.

*In arithmetic sequence I wanted to (know the number of terms) using these details:

$s_n = 270$

$d = 1$

$a_1 = 4$

I thought about it and used the formula: $s_n = {n\over 2}(2a(n-1)d)$
then I substitute

$270 = {n\over 2} (2(4)+(n-1)(1))$

$270 = {n\over 2}(8+n-1)$

$270 = {7n+n^2\over 2}$

$540 = 7n+n^2$ # I dont think this works ?

Best Answer

Solve the quadratic equation; and choose the positive value for n you obtain as solution.

Edit:

$(n-20)(n+27)=0\implies n-20=0 \text{ or } n+27=0\implies n=20\text{ or }-27$, now pick the positive one of it.

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