Question

HELP PLEASE

Find n for a series for which a1 = 30, d = -4, and Sn = -210.
a.
-21
c.
-10
b.
21
d.
10

Answer 1

To find the value of n in the given arithmetic sequence with a1 = 30 and d = -4, we can use the formula for the nth term and the sum of the first n terms. By solving the resulting quadratic equation, we find that n is approximately -10.

Step-by-step explanation:

To find n for the given series, we can use the formula for the nth term of an arithmetic sequence:

an = a1 + (n-1)d

Plugging in the given values a1 = 30 and d = -4, we have:

an = 30 + (n-1)(-4)

an = 30 - 4n + 4

Now, we need to find the value of n that makes the sum of the first n terms, Sn, equal to -210. The formula for the sum of the first n terms of an arithmetic sequence is:

Sn = n/2(a1 + an)

Plugging in the given values Sn = -210, a1 = 30, d = -4, and an = 30 + (n-1)(-4), we can solve for n:

-210 = n/2(30 + 30 - 4n)

-210 = n/2(60 - 4n)

-210 = n(60 - 4n)

-210 = 60n - 4n^2

4n^2 - 60n - 210 = 0

After solving this quadratic equation, we find that the value of n is approximately -10.

Answer 2

`\bf \qquad \qquad \textit{sum of a finite arithmetic sequence}\\\\ S_n=\cfrac{n}{2}(a_1+a_n)~ \begin{cases} n=n^(th)\ term\\ a_1=\textit{first term's value}\\ a_n=n^(th)~value\\ ----------\\ a_1=30\\ S_n=-210 \end{cases} \implies -210=\cfrac{n}{2}(30+a_n)\\\\\\ -420=n(30+a_n)\implies -420=30n+na_n \\\\\\ -420-30n=na_n\implies \boxed{\cfrac{-420-30n}{n}=a_n}\\\\ -------------------------------`


`\bf n^(th)\textit{ term of an arithmetic sequence}\\\\ a_n=a_1+(n-1)d\qquad \begin{cases} n=n^(th)\ term\\ a_1=\textit{first term's value}\\ d=\textit{common difference}\\ ----------\\ a_1=30\\ d=-4 \end{cases} \\\\\\ a_n=30+(n-1)(-4)\implies a_n=30-4n+4 \\\\\\ \boxed{a_n=34-4n}\\\\ -------------------------------`


`\bf \stackrel{a_n}{34-4n}~~=~~\stackrel{a_n}{\cfrac{-420-30n}{n}}\implies 34n-4n^2=-420-30n \\\\\\ 0=4n^2-64n-420\implies 0=4(n^2-16n-105) \\\\\\ 0=n^2-16n-105\implies 0=(n-21)(n+5)\implies n= \begin{cases} \boxed{21}\\ -5 \end{cases}`

since the term is just a term unit in the sequence, it cannot be a negative value, thus is not -5.