Question

find the 20th term of the A.P 80,75,70... calculted the number of terms required to make the sum equal to zero

Answer 1

Given: A.P 80,75,70 ... 
To find: 20th term of A.P and
             no. of terms required to make the sum 0

So, from A.P we have, 
a=80, n=20 and d=-5

∴ using the formula ,
an = a + (n-1)(d), we get
a20 = 80 + (20-1)(-5)
⇒a20 = 80 + (-95)
⇒a20 = -15

Also, using the formula
sn = n/2(2a+(n-1)(d)),we get
0 = n/2(2(80)+(n-1)(-5))
⇒0 = 165n - 5nsquare
⇒0 = 33-n
⇒n = 33

∴ The sum of 33 terms of the A.P will be 0

Answer 2

Given: A.P 80, 75, 70...

To find: The 20th term of the A.P. and the number of terms required to make the sum equal to zero.

Solution:

To calculate a term in an A.P., the following formula is used.

$n^{th} = a + (n-1)d$

Here, n is the term to be found, a is the first term and d is the common difference of the sequence. In the given sequence, the common difference is (-5).

$20^{th} = 80 + (20-1)(-5)$

       $= 80 + (19)(-5)$

       $= -15$

So, the 20th term of the A.P. is -15.

The sum of an A.P. is given by the formula.

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

Since the sum needs to be zero, it will be substituted in place of sum.

$0 = \frac{n}{2} [2(80) + (n-1)(-5)]$

$160n - 5n^{2} + 5n = 0$

$5n^{2} = 165 n$

$n = \frac{165}{5}$

$n = 33$

Therefore, the 20th term of the A.P. is -15 and the number of terms required to make the sum equal to zero is 33.