Question

The sum of the first three terms of an AP is 21 and their product is 231. Find the number.

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Answer 1

$\boxed{\textbf{\large{Explanation:}}}$

consider the 1st three terms of AP are ,

(a - d) , (a), (a + d)

▶the sum of 1st three terms of AP is 21

therefor,

→ ( a - d) + (a) + (a + d) = 21

→ a - d + a + a + d = 21

→ 3a = 21

→ a = 7

▶ their product is 231

→( a - d) (a) (a + d) = 231

put the value of a

→ (7 - d) (7) (7 + d) = 231

→ (7 - d) (7+ d) = 231 / 7

→ (7² - d² ) = 33

→ 49 - d² = 33

→ -d² = 33 - 49

→ - d² = - 16

→ d = 4

therefor, the terms are,

$\boxed{\textbf{\large{( a - d) = ( 7 - 4) = 3}}}$

$\boxed{\textbf{\large{(a ) = 7}}}$

$\boxed{\textbf{\large{(a + d) = ( 7 + 4 ) = 11}}}$

Answer 2

Answer:

3, 7 and 11

Step-by-step explanation:

Let the first three terms of AP be a, (a + d) and (a + 2d) respectively.

Given that,

Sum of three terms is 21.

⇒ a + a + d + a + 2d = 21

⇒ 3a + 3d = 21

⇒ 3( a + d) = 21

⇒ a + d = 7 .... (i)

⇒ a = 7 - d ... (ii)

Now, their product is 231.

⇒ a (a + d) (a + 2d) = 231

⇒ (7 - d) * 7 * (7 - d + 2d) = 231

⇒ 7(7 - d) (7 + d) = 231

⇒ 7 { 7² - d² } = 231

⇒ 7² - d² = $\sf \dfrac{231}{7}$

⇒ 49 - d² = 33

⇒ d² = 49 - 33

⇒ d² = 16

⇒ d = ± √16

⇒ d = ± 4

Taking the positive value, d = 4.

Substituting the value of d in (ii),

⇒ a = 7 - d

⇒ a = 7 - 4

⇒ a = 3

_______________________

Now, the terms of A.P are ;

• a = 3
• a + d = 3 + 4 = 7
• a + 2d = 3 + 2 * 4 = 11

Hence, the terms of A.P are 3, 7, 11....