If the square of the 7th term of an arithmetic progression with positi

Question

If the square of the 7th term of an arithmetic progression with positive common difference equals the product of the 3rd and 17th terms, then the ratio of the first term to the common difference is:

A. 2 : 3
B. 3 : 2
C. 3 : 4
D. 4 : 3
E. 5 : 3

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nick1816 · Answer

The nth term of arithmetic progression can be written as  = a+(n-1)d

a= first term; d = common difference

7th term of AP  = a+6d

3rd term of AP  = a+2d

17th term of AP  = a+16d

(a+6d)^2 = (a+2d)(a+16d)

a^2+12ad+36d^2 = a^2+18ad + 32d^2

6ad= 4d^2

Since d>0

6a = 4d

\frac{a}{d} = \frac{2}{3}

yashikaaggarwal · Answer

(A+6D)^2=(A+2D)(A+16D) given
A^2+36D^2+12AD=A^2+32D^2+18AD
4D^2=6AD
2D=3A
A/D=2/3

Answer is A

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GMATWhizTeam · Answer

Solution 
 • Let us assume that  a_1, d,  and  a_n  denotes first term, common difference and nth term of the given A.P. respectively.
 o Where n is a positive integer. 
•  d > 0 
• We need to find  \frac{a_1}{d} 
• Now, according to the questions
   (a_7)^2 = a_3*a_{17} 
  ⟹ (a_1+ 6d)^2 = (a_1 + 2d)(a_1+16d) 
  ⟹(a_1)^2 + 12a_1*d + 36d^2 = (a_1)^2 + 18a_1*d + 32d^2 
  ⟹ 6a_1*d – 4d^2 = 0 
  ⟹ d(6a_1 – 4d) = 0  
  ⟹ \frac{a_1 }{ d} = \frac{4}{6 }= \frac{2}{3}   
Thus, the correct answer is  Option A.

sambitspm · Answer

See the attachment.

Kinshook · Answer

Asked: If the square of the 7th term of an arithmetic progression with positive common difference equals the product of the 3rd and 17th terms, then the ratio of the first term to the common difference is:

Let a be first term and d be common difference of the arithmetic progression.

(a + 6d)^2 = (a+2d)(a+16d)
a^2 + 12ad + 36d^2 = a^2 + 18ad + 32d^2
6ad = 4d^2; 6a = 4d; a:d = 4:6 = 2:3

IMO A

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If the square of the 7th term of an arithmetic progression with positi

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