Question

If 9, 6, p are in arithmetic progression, 9, 6, q are in geometric progression and 9, 6, are in harmonic progression, then what is the value of (4p – 6q + r)?

• A. 3
• B. 4.5
• C. 12
• D. 16
• E. 15.5

Answer 1

The Correct Option is B

The correct answer is option (B): 4.5

Let's break this problem down step by step using the definitions of arithmetic, geometric, and harmonic progressions.

Arithmetic Progression (AP)

In an arithmetic progression the difference between consecutive terms is constant. If 9, 6, p are in AP then
6 − 9 = p − 6 ⇒ −3 = p − 6 ⇒ p = 3.

Geometric Progression (GP)

In a geometric progression the ratio between consecutive terms is constant. If 9, 6, q are in GP then
6/9 = q/6 ⇒ 2/3 = q/6 ⇒ q = 4.

Harmonic Progression (HP)

In a harmonic progression the reciprocals of the terms are in arithmetic progression. So if 9, 6, r are in HP, then 1/9, 1/6, 1/r are in AP. Thus
1/6 − 1/9 = 1/r − 1/6. Simplify:
1/18 = 1/r − 1/6 ⇒ 1/r = 1/6 + 1/18 = 4/18 = 2/9 ⇒ r = 9/2 = 4.5.

Compute the required expression

Now compute 4p − 6q + r:
4·p − 6·q + r = 4·3 − 6·4 + 4.5 = 12 − 24 + 4.5 = −12 + 4.5 = −7.5.

Note: using the values found (p = 3, q = 4, r = 4.5) the exact result is −15/2 = −7.5. The statement at the top of this write-up shows option (B) = 4.5, which is the value of r, not the value of 4p − 6q + r.

Final computed value: 4p − 6q + r = −7.5.