Question

The first term of an arithmetic progression is 3000 and the tenth term is 1200. (i) Find the sum of the first 20 terms of the progression. (ii) After how many terms does the sum of the progression become negative?​

Answer 1

Given:

a = 3000

$\displaystyle t_1_0$ = 1200

To find:

1) $\displaystyle S_2_0$ =?

2) n =?

Step-by-step explanation:

The general term of an AP is given by

$\displaystyle t_n =$ a + ( n -1 ) d        

The tenth term of an AP is 1200.

a = 3000, n = 10

Put these values in above equation

1200 = 3000 + ( 10 -1) d

1200 = 3000 + 9d

⇒ d = -200

1)

The sum of first n terms of an AP is given by

$\displaystyle S_n = \frac{n}{2} \ [2a + (n-1)d \ ]$                  ...(1)

The sum of the first 20 terms of an AP is given by

$\displaystyle S_2_0 = \frac{20}{2} \ [2( 3000)+ (20-1)(-200) \ ]$

$\displaystyle\mathbf{ S_2_0}$ = -22000

2)

Let n be the no of terms having a sum of the progression become negative.

i.e. $\displaystyle S_n$ = 0

from above equation (1)

$\displaystyle \frac{n}{2} \ [2a + (n-1)d \ ] = 0$

$\displaystyle \frac{n}{2} \ [2(3000) + (n-1)(-200) \ ] \ = 0$

⇒ n = 31

∴ After 31 th term, the sum of progression becomes negative.