The quotient when L.C.M. is divided by the H.C.F. of a G.P. with first term 'a' and common ratio 'r' is:

Options

A. `r^(n -1)`
B. `r^(n)`
C. `a^(-1)r^(n-2)`
D. `(r^n - 1)`

Answer

To find the quotient when the L.C.M. (Least Common Multiple) is divided by the H.C.F. (Highest Common Factor) of a geometric progression (G.P.) with first term 'a' and common ratio 'r', we can follow these steps:

### Step 1: Understand the terms of the G.P.
A geometric progression with first term 'a' and common ratio 'r' can be expressed as:
- First term: \( a \)
- Second term: \( ar \)
- Third term: \( ar^2 \)
- ...
- \( n \)-th term: \( ar^{n-1} \)

### Step 2: Identify the L.C.M. of the G.P.
The L.C.M. of the terms in a G.P. can be determined by taking the highest power of each term. For the terms \( a, ar, ar^2, \ldots, ar^{n-1} \), the L.C.M. is:
\[ \text{L.C.M.} = ar^{n-1} \]
This is because \( ar^{n-1} \) is the largest term in the progression.

### Step 3: Identify the H.C.F. of the G.P.
The H.C.F. of the terms in a G.P. is the common factor present in all terms. Here, 'a' is the common factor in all terms:
\[ \text{H.C.F.} = a \]

### Step 4: Calculate the quotient of L.C.M. divided by H.C.F.
Now, we can find the quotient when L.C.M. is divided by H.C.F.:
\[ \text{Quotient} = \frac{\text{L.C.M.}}{\text{H.C.F.}} = \frac{ar^{n-1}}{a} \]
When we simplify this, we get:
\[ \text{Quotient} = r^{n-1} \]

### Conclusion
Thus, the quotient when the L.C.M. is divided by the H.C.F. of a G.P. with first term 'a' and common ratio 'r' is:
\[ r^{n-1} \]

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