Question

10th and 20th term of a H.P are 1/5and 1/10 respectively.find the 40th term


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Answer 1

Answer:-

Given:

10th term of a HP = 1/5

20th term = 1/10

We know that,

HP is inverse of AP.

So,

10th term of an AP = 5

20th term = 10

We know,

nth term of an AP (aₙ) = a + (n - 1)d

So,

★ a + (10 - 1)d = 5

⟹ a + 9d = 5 -- equation (1).

★ a + (20 - 1)d = 10

⟹ a + 19d = 10 -- equation (2).

Subtract equation (1) from equation (2).

⟹ a + 19d - (a + 9d) = 10 - 5

⟹ a + 19d - a - 9d = 5

⟹ 10d = 5

⟹ d = 5/10

⟹ d = 1/2

Substitute the value of d in equation (1)

⟹ a + 9(1/2) = 5

⟹ a = 5 - 9/2

⟹ a = (10 - 9)/2

⟹ a = 1/2

Now,

40th term of the AP = 1/2 + (40 - 1)(1/2)

⟹ a₄₀ = 1/2 + 39/2

⟹ a₄₀ = (1 + 39)/2

⟹ a₄₀ = 40/2

⟹ a₄₀ = 20

⟹ 40th term of HP = 1/20

∴ The 40th term of the required HP is 1/20.

Answer 2

Solution:

Given:

T10 = 1/5

T20 = 1/10

T10 = a + (n - 1) d

1/5 = a + (10 - 1)d

1/5 = a + 9d ... Eq.1

T20 = a + (n - 1) d

1/10 = a + (20 - 1)d

1/10 = a + 19d ... Eq.2

Subtract Eq.1 and Eq.2 to solve for d

1/5 = a + 9d
-
1/10 = a + 19d

d = -1/100

Solve for a, use Eq.1

1/5 = a + 9(-1/100)

a = 29/100

First term, a = 29/100

Common difference, d = -1/100

T40 = 29/100 + (40 - 1) -1/100

T40 = 29/100 + 39(-1/100)

= 29/100 - 39/100

= -1/10

Therefore, T40 = -1/10

Hope this will be helpful to you.