If the 8th term of an AP is 31 and its 15th term is 16 more than the 1

1.	If the 8th term of an AP is 31 and its 15th term is 16 more than the 11th term, find the AP.
Answer» Let first term of AP is a and common difference of AP is d. We know that, n^th term of AP is an = a + (n – 1) d. Given that, 8^th term of AP is a₈ = 31 ⇒ a + (8 – 1) d = 31 ⇒ a + 7d = 31 … (1) Also, Given that, 15th term of AP is 16 more than the 11^th term of AP. i.e., a₁₅ = 16 + a₁₁ ⇒ a + 14d = 16 + a + 10d ⇒ 14d − 10d = 16 ⇒ d = \(\frac{16}{4}\) = 4. From equation (1), a + 7 × 4 = 31 (∵d = 4) ⇒ a = 31 – 28 = 3. Hence, The first term of AP is a₁ = a = 3. The second term of AP is a₂ = a + d = 3 + 4 = 7. The third term of AP is a₃ = a + 2d = 3 + 8 = 11. Hence, The required AP is 3, 7, 11, ……… .

If the 8th term of an AP is 31 and its 15th term is 16 more than the 11th term, find the AP.

Answer»

Let first term of AP is a and common difference of AP is d.

We know that,

n^th term of AP is an = a + (n – 1) d.

Given that,

8^th term of AP is a₈ = 31

⇒ a + (8 – 1) d = 31

⇒ a + 7d = 31 … (1)

Also,

Given that,

15th term of AP is 16 more than the 11^th term of AP.

i.e., a₁₅ = 16 + a₁₁

⇒ a + 14d = 16 + a + 10d

⇒ 14d − 10d = 16

⇒ d = \(\frac{16}{4}\) = 4.

From equation (1),

a + 7 × 4 = 31

(∵d = 4)

⇒ a = 31 – 28 = 3.

Hence,

The first term of AP is a₁ = a = 3.

The second term of AP is a₂ = a + d

= 3 + 4 = 7.

The third term of AP is a₃ = a + 2d

= 3 + 8

= 11.

Hence,

The required AP is 3, 7, 11, ……… .

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