Question

The sum of first six terms of an AP is 42. The ratio of its 10h term to its 30 term
Calculate the first and the 30th term of the AP.
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Answer 1

Given:

• Sum of first six terms of an AP is 42.
• Ratio of 10th to its 30th term 1:3.

To find:

• 30th term of AP?

Solution:

★ According to the Question:

• Ratio of 10th to its 30th term 1:3.

→ (a + 9d)/(a + 29d) = 1/3

→ 3(a + 9d) = a + 29d

→ 3a + 27d = a + 29d

→ 3a - a = 29d - 27d

→ 2a = 2d

→ 2a - 2d = 0 ....eq (1)

Also,

• Sum of first six terms of an AP is 42.

→ S₆ = n/2 (2a + (n - 1)d)

→ 42 = 6/2 (2a + (6 - 1)d)

→ 42 = 3(2a + 5d)

→ 42/3 = 2a + 5d

→ 14 = 2a + 5d

→ 2a + 5d = 14 ....eq (2)

Now, Solving equation (1) and (2), we get,

• a = 2 and d = 2

Now, Finding 30th term of AP,

→ a₃₀ = a + 29d

→ a₃₀ = 2 + 29 × 2

→ a₃₀ = 2 + 58

→ a₃₀ = 60

∴ Hence, 30th term of AP is 60.

Answer 2

Answer:

Given:

Sum of first six terms of an AP

is 42.

Ratio of 10th to its 30th term 1:3.

To find:

• 30th term of AP?

Solution:

* According to the Question:

• Ratio of 10th to its 30th term 1:3.

(a + 9d)/(a + 29d) = 1/3

3(a + 9d) = a + 29d

- 3a + 27d = a + 29d

- 3a - a = 29d 27d

- 2a = 2d

- 2a - 2d = 0....eq (1)

Also,

Sum of first six terms of an AP is 40

+ S6 = n/2 (2a + (n - 1)d) =

42 = 6/2 (2a + (6 - 1)d)

42 = 3(2a + 5d)

+ 42/3 = 2a + 5d

- 14 = 2a + 5d

- 2a + 5d = 14 ...eq (2)

Now, Solving equation (1) and (2), we get,

a = 2 and d = 2

Now, Finding 30th term of AP,

→ a3o = a + 29d

+ a30 = 2 + 29 x 2

+ a30 = 2 + 58

a30 60

.. Hence, 30th term of AP is 60.