Question

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$\implies$ The sum of three terms in an A.P. is 21 and their product is 231. Find the numbers.

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Kindly:-

$\bullet{\leadsto}$ Do explain properly why to take a, a-d and a+d instead of a, a+d and a+2d.
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Answer 1

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⇒ The sum of three terms in an A.P. is 21 and their product is 231. Find the numbers.

Given:-

• The sum of 3 numbers in A.P is 21 and the product is 231.

To find:-

• The 3 numbers

$\huge\underline{\overline{\mid{\bold{\purple{\mathrm{Solution:- }}\mid}}}}$

• Let the first term be a, and the common difference be d (as it is an arithmetic progression)

$\underline{ \boxed{ \sf \: Let \: (a-d),a,(a+d) \: are \: 3 \: numbers \: in \: A.P}}$

$\red➦a - d + a + a + d = 21$

$\pink➦3a = 21$

$\orange➦a = \frac{21}{3}$

$\purple➦a = 7$

• Therefore the value of a is 7

$\blue \implies \: (a - d)a(a + d) = 231$

$\green \implies \: a( {a}^{2} - {d}^{2} ) = 231$

${ \color{aqua} {\implies}} \: 7(49 - {d}^{2} ) = 231$

${ \color{maroon}{ \implies}}49 - {d}^{2} = \frac{231}{7}$

${ \color{gold}{ \implies}}49 - {d}^{2} = 33$

${ \color{blue}{ \implies}}{d}^{2} = 49 - 33$

${ \color{navy}{ \implies}} {d}^{2} = 16$

$\: \: \: \: \: \: { \color{yellow}{ \implies}}d = \sqrt{16}$

${ \color{green}{ \implies}}d = 4$

⇰a-d =7-4=3

⇰a+d=7+4=11

⇰a=7

$\boxed{ \underline{ \mathcal{ \green{Therefore \: the \: numbers \: are \: 3,11,7}}}}$

Answer 2

EXPLANATION.

Sum of three terms in an A.P. = 21.

Their products = 231.

As we know that,

Three terms of an A.P. = a - d, a, a + d.

⇒ a - d + a + a + d = 21.

⇒ 3a = 21.

⇒ a = 7.

⇒ (a - d)(a)(a + d) = 231.

As we know that,

Formula of :

⇒ x² - y² = (x + y)(x - y).

Using this formula in equation, we get.

⇒ (a - d)(a + d)(a) = 231.

⇒ (a² - d²)(a) = 231.

Put the value of a = 7 in equation, we get.

⇒ [(7)² - d²](7) = 231.

⇒ [49 - d²](7) = 231.

⇒ 343 - 7d² = 231.

⇒ -7d² = 231 - 343.

⇒ -7d² = -112.

⇒ 7d² = 112.

⇒ d² = 16.

⇒ d = √4.

⇒ d = ± 4.

First term of an A.P. = a = 7.

Common difference = d = b - a = 4.

Three numbers are,

⇒ (a - d), a, (a + d).

⇒ (7 - 4), 7, (7 + 4).

⇒ 3, 7, 11.

First term of an A.P. = a = 7.

Common difference = d = b - a = -4.

Three numbers are,

⇒ (a - d), a, (a + d).

⇒ [7 - (-4)], 7, [7 + (-4)].

⇒ [7 + 4], 7, [7 - 4].

⇒ 11, 7, 3.

                                                                                                                     

MORE INFORMATION.

(1) = Arithmetic progression (A.P.)

If a is the first term and d is the common difference then A.P. can written as,

a + (a + d) + (a + 2d) + ,,,

(2) = General term of an A.P.

General term (nth term) of an A.P. is given by,

Tₙ = a + (n - 1)d.

(3) = Sum of n terms of an A.P.

Sₙ = n/2 [2a + (n - 1)d]  Or  Sₙ = n/2[a + Tₙ].

(1) = If sum of n terms Sₙ is given then general term Tₙ = Sₙ - Sₙ₋₁ Where (Sₙ₋₁) is sum of (n - 1) terms of A.P.

(4) = Arithmetic mean (A.M.)

If A is the A.M. between two given numbers a and b, then

A = a + b/2 ⇒ 2A = a + b.