Q. 3 B3.6( 9 Votes )

# Divide 20 into four parts which are in A.P. such that the ratio of the product of the first and fourth is to the product of the second and third is 2 : 3.

Answer :

Let the four parts which are in AP are

(a – 3d), (a – d), (a + d), (a + 3d)

According to question,

The sum of these four parts = 20

⇒(a – 3d) + (a – d) + (a + d) + (a + 3d) = 20

⇒ 4a = 20

⇒ a = 5 …(i)

Now, it is also given that

product of the first and fourth : product of the second and third = 2 : 3

i.e. (a – 3d) × (a + 3d) : (a – d) × (a + d) = 2 : 3

[∵(a – b)(a + b) = a^{2} – b^{2} ]

⇒ 3(a^{2} – 9d^{2}) = 2(a^{2} – d^{2})

⇒ 3a^{2} – 27d^{2} = 2a^{2} – 2d^{2}

⇒ 3a^{2} – 2a^{2} = – 2d^{2} + 27d^{2}

⇒ (5)^{2} = – 2d^{2} + 27d^{2} [from (i)]

⇒ 25 = 25d^{2}

⇒ 1 = d^{2}

⇒ d = ±1

Case I: if d = 1 and a = 5

a – 3d = 5 – 3(1) = 5 – 3 = 2

a – d = 5 – 1 = 4

a + d = 5 + 1 = 6

a + 3d = 5 + 3(1) = 5 + 3 = 8

Hence, the four parts are

2, 4, 6, 8

Case II: if d = – 1 and a = 5

a – 3d = 5 – 3( – 1) = 5 + 3 = 8

a – d = 5 – ( – 1) = 5 + 1 = 6

a + d = 5 + ( – 1) = 5 – 1 = 4

a + 3d = 5 + 3( – 1) = 5 – 3 = 2

Hence, the four parts are

8, 4, 6, 2

Rate this question :

The sum of first n terms of an A.P. is 5n – n^{2}. Find the nth term of this A.P.

If the sum of the first n terms of an A.P. is 4n – n^{2}, which is the first term? What is the sum of first two terms? What is the second term? Similarly, find the third, the tenth and the nth terms.

The sum of the first 7 terms of an A.P. is 63 and the sum of its next 7 terms is 161. Find the 28^{th} term of this A.P.

The sum of first q terms of an A.P. is 63q – 3q^{2}. If its pth term is-60, find the value of p. Also, find the 11^{th} term of this A.P.

Find the number of terms of the A.P. If 1 is added to each term of this A.P., then find the sum of all terms of the A.P. thus obtained.

RD Sharma - MathematicsThe sum of first m terms of an A.P. is 4 m^{2} - m. If its nth term is 107, find the value of n. Also, find the 21^{st} term of this A.P.

The sum of first n terms of an A.P. is 3n^{2} + 4n. Find the 25^{th} term of this A.P.

The sum of first seven terms of an A.P. is 182. If its 4^{th} and the 17^{th} terms are in the ratio 1:5, find the A.P.

Sum of the first 14 terms of an A.P. is 1505 and its first term is 10. Find its 25th term.

RD Sharma - MathematicsIf demotes the sum of first n terms of an A.P., prove that S_{12} = 3 (s_{8} – S_{4}) .