Question

Directions for questions 67 to 70: Seven integers A, B, C, D, E, F and G are to be arranged in increasing order such that:
(i) The first four numbers, in this increasing order, are in arithmetic progression (A.P.).
(ii) The last four numbers, in this increasing order, are in geometric progression (G.P.).
(iii) There is exactly one number between E and G, in the increasing order.
(iv) There is no number between A and B, in the increasing order.
(v) D is the smallest number and E is the greatest number.
(vi) \[ \frac{A}{D}=\frac{G}{C}=\frac{F}{A}>1 \]
(vii) E = 960

69. The common difference in the A.P. is:

• A. 20
• B. 22
• C. 25
• D. 30

Hint:

Find the common difference in ratio units first (it comes out to 1), then scale by the same factor used to make E equal 960.

Answer 1

The Correct Option is D

A quicker route to the same seven numbers is to first solve the puzzle in "ratio units," ignoring the actual size of D, and only bring in E = 960 at the very end.

Assume for a moment that D = 1 (any positive starting value works, since the whole picture just scales up or down together). The seven numbers sit in the order D, C, B, A, G, F, E, with D and E at the two ends, and A/D = G/C = F/A all equal to one number k, where A is also the fourth term of the A.P. and the first term of the G.P.

1. With D = 1 and A.P. common difference d, the A.P. reads 1, 1+d, 1+2d, 1+3d, so A = 1+3d.
2. The G.P. starts at A with ratio r, so G = Ar, F = Ar^2, E = Ar^3, and F/A = r^2 must equal A/D = A.
3. Matching A/D with G/C also ties d and r together, and solving the two relations at once (exactly as in the direct method) gives d = 1 and r = 2, so A = 1+3(1) = 4 and k = r^2 = 4.

So in ratio units: D = 1, C = 2, B = 3, A = 4, G = 8, F = 16, E = 32.

Before scaling up, it is worth checking that these seven ratio-unit numbers already satisfy every condition in the directions, since scaling by a positive constant never breaks an order relation, a common difference/ratio relation, or a ratio like A/D:

• Increasing order: 1, 2, 3, 4, 8, 16, 32, which is D, C, B, A, G, F, E in that order.
• First four (D, C, B, A) = 1, 2, 3, 4: common difference 1, a genuine A.P.
• Last four (A, G, F, E) = 4, 8, 16, 32: common ratio 2, a genuine G.P.
• Exactly one number, F = 16, sits between G = 8 and E = 32.
• No number sits between A = 4 and B = 3, they are next-door values.
• A/D = 4/1 = 4, G/C = 8/2 = 4, F/A = 16/4 = 4: all equal and above 1, exactly as condition (vi) needs.

Every condition holds in ratio units, so scaling the whole picture by a single positive number keeps every condition true. Now scale everything so E matches the real value 960. Since E = 32 in ratio units and E = 960 in real units, the scale factor is 960/32 = 30. Multiplying every ratio-unit value by 30 gives:

\[ D=30,\ C=60,\ B=90,\ A=120,\ G=240,\ F=480,\ E=960 \]

These match the direct method exactly, and since every condition already held true in ratio units, it automatically still holds true after scaling by 30.

In ratio units the A.P. is 1, 2, 3, 4, so its common difference is 1 unit. Scaling by 30 (the factor that makes E match 960) turns that common difference of 1 into a real common difference of 30.

Let's summarize:

• The ratio-unit A.P. 1, 2, 3, 4 has common difference 1.
• Scaling everything by 30 turns the A.P. into 30, 60, 90, 120, with common difference 30.
• No other option (20, 22 or 25) fits this scaled A.P.

\[ \boxed{d=30} \]