Answer:
The value of n is 21
Step-by-step explanation:
The series is 19 , 15 , 11 , ............
It has a constant difference between each two consecutive terms, so it is an arithmetic series.
The rule of the sum of n terms in the arithmetic series is:
[tex]S_{n}=\frac{n}{2}[2a+(n-1)d][/tex]
Where a is the first term of the series , d is the difference between each two consecutive terms and n is the number of the terms in this series
So from the series above:
a = 19 , d = 15 - 19 = -4 ( it must be second term - first term ) , [tex]S_{n}=-441[/tex]
[tex]-441=\frac{n}{2}[2(19)+(n-1)(-4)][/tex]
[tex]-441=\frac{n}{2}[38+(-4n+4)][/tex]
[tex]-441=\frac{n}{2}[38-4n+4][/tex]
[tex]-441=\frac{n}{2}[42-4n][/tex]
[tex]-441=21n-2n^{2}[/tex]
[tex]2n^{2}-21n-441=0[/tex] ⇒Use factorization
(2n + 21) (n - 21) = 0
2n + 21 = 0 ⇒ 2n = -21 ⇒ n = -21/2⇒Not accepted n must be positive integer
n - 21 = 0⇒ n = 21
∴-441 is the sum of 21 terms in this series
