Question

If $a, b, c$ are in geometric progression and $a + b + c = 21$, $abc = 216$, what is the value of $b$? 
 

• A. 6
• B. 7
• C. 8
• D. 9

Hint:

For GP problems, use the common ratio and given conditions to form equations, then test options for quick confirmation.

Answer 1

The Correct Option is A

- Step 1: Since $a, b, c$ are in GP, $b^2 = ac$. 
Let the common ratio be $r$, so $b = ar$, $c = ar^2$. 
- Step 2: Given $a + b + c = 21$, so $a + ar + ar^2 = a(1 + r + r^2) = 21$. 
- Step 3: Given $abc = 216$, so $a \cdot ar \cdot ar^2 = a^3 r^3 = 216$. 
- Step 4: From $b^2 = ac$ and $c = br$, we have $b = ar$, $c = ar^2$, 
so $abc = a \cdot ar \cdot ar^2 = a^3 r^3 = 216$, and $b^3 = a^2 r^3 \cdot r = 216 \cdot r$. 
- Step 5: Test $b = 6$: 
Then $b^3 = 216$, so $a^3 r^3 = 216$, $a^3 r^3 = b^3$, $r = 1$. 
If $r = 1$, $a = b = c = 6$. Check: $6 + 6 + 6 = 18 \neq 21$. 
Try $r \neq 1$, solve numerically via options. 
$b = 6$, $a = \dfrac{6}{r}$, $c = 6r$, then $6/r + 6 + 6r = 21$, $6r^2 - 15r + 6 = 0$, roots give $b = 6$. 
- Step 6: Option (a) is 6, which matches.