Question

In an A.P, ten times the tenth term is equal to thirty times it's 30th term

Answer 1

Correct Question;

In an A.P ten times of its tenth term is equal to thirty times of its 30 th term .Find its 40th term.

Theory :

General term of an AP

$\sf \: a_{n} = a + (n - 1)d$

Where ,d = common difference

n = no of terms

and a = first term

Solution :

Let the first term of an AP be 'a' and common difference be 'd'.

$\sf \: a_{10} = a + (10 - 1)d$

$\implies \sf \: a_{n} = a + 9d$

$\sf \: a_{30} = a + (30 - 1)d$

$\implies\sf \: a_{30} = a + 29d$

According to the question:

$\sf \: 10a_{10} = 30a_{30}$

$\implies \sf \: 10a + 90d = 30a + 870d$

$\implies \sf30a - 10a + 870d - 90d = 0$

$\sf \implies20a + 780d = 0...(1)$

Now , 40th term

$\sf \: a_{40} = a + (40- 1)d$

$\sf \: a_{40} = a + 39d$

Multiply both sides by 20

$\implies \sf \:20 \times a_{40} = 20 \times (a + 39d)$

$\implies \sf \:20 \times a_{40} = 20a + 780d$

Form equation (1) [ 20a+780d = 0]

$\implies \sf \:20 \times a_{40} = 0$

$\implies \sf \: a_{40} =0$

More About Arithmetic Progression:

Sum of n terms of an AP

$\sf \:20 s_{10} = \dfrac{n}{2} (2a + (n - 1)d)$

Answer 2

QUESTION

In an A.P ten times of its tenth term is equal to thirty times of its 30th term. Find its 40th term.

ANSWER

Ten times of its tenth term is equal to thirty times of its 30th term

$\implies 10(a+9d) =30(a+29d)$

$\implies a+9d = 3(a+29d)$

$\implies a +9d = 3a + 87d$

$\implies a-3a=87d-9d$

$\implies -2a=78d$

$\implies a = \dfrac{-78d}{2}$

$\implies a=-39d$

The 40th term

$= a+39d\\= -39d+39d\\=0$

$\boxed{\boxed{\bold{Therefore, \ the \ 40th \ term \ is \ 0}}}}}$