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

1.	The sum of three terms in an A.P. is 21 and their product is 231. Find the numbers.___________________________________Kindly:- Do explain properly why to take a, a-d and a+d instead of a, a+d and a+2d. Quality answer required!
Answer» 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.

The sum of three terms in an A.P. is 21 and their product is 231. Find the numbers.___________________________________Kindly:- Do explain properly why to take a, a-d and a+d instead of a, a+d and a+2d. Quality answer required!

Answer»

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.