# What is the common difference of a 43-term arithmetic sequence where the first term is -13 and the sum is 9,374?

[latex]\bf \textit{sum of a finite arithmetic sequence}\\\\ S_n=\cfrac{n}{2}(a_1+a_n)\qquad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\ a_n=\textit{value of the }n^{th}\ term\\ ----------\\ n=43\\ a_1=-13\\ S_n=9374 \end{cases} \\\\\\ 9374=\cfrac{43}{2}(-13+a_{43})\implies 18748=43(-13+a_{43}) \\\\\\ \cfrac{18748}{43}=-13+a_{43}\implies 436+13=a_{43}\implies \boxed{449=a_{43}}[/latex] so.. .we know the first term is -13, and the 43rd term's value is 449...so...this is an arithmetic sequence, thus, in order for us to get from -13 to 449 in 43 hops, we had to add "d" the common difference, 42 times. so -13, -13 + d, (-13+d)+d, (-13+d+d)+d, ..... and so on. but in short d + d + d + ... 42 times is just 42d so, we know that [latex]\bf \begin{array}{llll} -13+42d=&449\\ \ \uparrow &\ \uparrow\\ a_1&a_{43} \end{array}\implies 42d=462\implies d=\cfrac{462}{42}\implies \boxed{d=11}[/latex]