​What is the value of k in the following Arithmetic progression?

15 + 13 + 11 + 9 + ..... + k = –105

Correct option is C

​Given:

First term a=15

Common difference $d=-2$

Sum of terms $S$_n=−105

Formula Used:

The sum of the first $n$ terms of an arithmetic progression is given by the formula:

S_n= $$\frac{n}{2}$$​(2a+(n−1)⋅d) 

Solution:

-105= $$\frac{n}{2}$$​​(30 +(n−1)⋅-2) 

−105=​$$\frac{n}{2}$$​(32−2n)

n²-16n -105= 0

Now, solve the quadratic equation using the quadratic formula:

$$n = \frac{-(-16) \pm \sqrt{(-16)^2 - 4(1)(-105)}}{2(1)} \\ \ \\ n = \frac{16 \pm \sqrt{256 + 420}}{2} \\ \ \\ n = \frac{16 \pm \sqrt{676}}{2} \\ \ \\ n = \frac{16 \pm 26}{2}$$

n = $$\frac{42}{2}=21$$ (valid solution)

n= $$\frac{-10}{2}=-5$$ (not valid since n must be positive)

So, n = 21

k=a+(n−1)⋅d

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