Question

The first term of an ap is 3 ,the sum of the first 5 terms is 1 11 th of the sum of next 5terms.Find the 20th term.

Answer 1

Solution:

The first term of the AP is given 3.

Let the common difference is d.

Then the AP can be written as

3, 3 + d, 3 + 2d, 3 + 3d, ...

Sum of the first 5 terms

= 3 + 3 + d + 3 + 2d + 3 + 3d + 3 + 4d

= 15 + 10d

and the sum of the next 5 terms

= 3 + 5d + 3 + 6d + 3 + 7d + 3 + 8d + 3 + 9d

= 15 + 35d

By the given condition,

sum of first 5 terms = 1/11 * sum of the next 5 terms

Then 15 + 10d = 1/11 * (15 + 35d)

or, 11 (15 + 10d) = 15 + 35d

or, 165 + 110d = 15 + 35d

or, 110d - 35d = 15 - 165

or, 75d = - 150

or, d = - 2

So the common difference, d = - 2

Therefore, the 20th term is

= 3 + (20 - 1) * (- 2)

= 3 - 38

= - 35

Answer 2

Answer:

Step-by-step explanation:

The first term of the AP is given as  3.

Let the common difference of the series be d.

Then the AP can be written as

3, 3 + d, 3 + 2d, 3 + 3d, ......

Since, it is being given that sum of first 5 terms is 1/11 th of the sum of the next 5 terms.

The sum of first five term is

= 3 + 3 + d + 3 + 2d + 3 + 3d + 3 + 4d

= 15 + 10d

and the sum of the next 5 terms is given by

= 3 + 5d + 3 + 6d + 3 + 7d + 3 + 8d + 3 + 9d

= 15 + 35d

Since the  given condition is;

sum of first 5 terms = 1/11 * sum of the next 5 terms

$15 + 10 \times d = 1/11 \times  (15 + 35d)$Then,

$or, 11  \times(15 + 10d) = 15 + 35d$

$or, 165 + 110 \times d = 15 + 35 \times d$

$or, 75 \times d = - 150$

or, d = - 2

So the common difference, d = - 2

Therefore, the 20th term is

$= 3 +  (20 - 1 )  \times (- 2)$

= 3 - 38

= - 35

So, the 20 th term of the given AP wil be -35