© All Rights reserved @ LearnWithDash
Step-by-Step Detailed Solution
Step 1: Express the fourth term of the GP
In a geometric progression (GP) with first term a and common ratio r = \frac{1}{m} , the n th term is given by T_n = a\,r^{n-1} . For the fourth term, we have:
T_4 = a \, r^3 = 500.
Hence,
a = \frac{500}{r^3}.
Step 2: Write down the sum of the first n terms
The sum of the first n terms of a GP is:
S_n = \frac{a\,(1 - r^n)}{1 - r}, \quad \text{for } r \neq 1.
Step 3: Use the condition S_6 > S_5 + 1
We have
S_6 = \frac{a\,(1 - r^6)}{1 - r}
\quad\text{and}\quad
S_5 = \frac{a\,(1 - r^5)}{1 - r}.
From the inequality:
S_6 > S_5 + 1,
we can write:
\frac{a\,(1 - r^6)}{1 - r} - \frac{a\,(1 - r^5)}{1 - r} > 1
\quad\Longrightarrow\quad
a \, r^5 > 1.
Substituting r = \frac{1}{m} and a = \frac{500}{r^3} = 500 m^3 , we get:
a\,r^5 = 500\,m^3 \,\left(\frac{1}{m}\right)^5 = 500\,\frac{1}{m^2} > 1
\quad\Longrightarrow\quad
\frac{500}{m^2} > 1
\quad\Longrightarrow\quad
500 > m^2
\quad\Longrightarrow\quad
m^2 < 500.
Hence,
m < \sqrt{500} \approx 22.36.
Since m \in \mathbb{N} , we get
m \le 22.
Step 4: Use the condition S_7 < S_6 + \tfrac{1}{2}
Similarly,
S_7 = \frac{a\,(1 - r^7)}{1 - r}
\quad\text{and}\quad
S_6 = \frac{a\,(1 - r^6)}{1 - r}.
From
S_7 < S_6 + \frac{1}{2},
we isolate the last term difference:
S_7 - S_6 = a\,r^6 < \frac{1}{2}.
Substituting a = \frac{500}{r^3} and r = \frac{1}{m} , note that
a \, r^6 = \frac{500}{\left(\frac{1}{m}\right)^3} \cdot \left(\frac{1}{m}\right)^6
= 500\,m^3 \,\frac{1}{m^6} = 500\,\frac{1}{m^3}.
So
500\,\frac{1}{m^3} < \frac{1}{2}
\quad\Longrightarrow\quad
\frac{1}{m^3} < \frac{1}{1000}
\quad\Longrightarrow\quad
m^3 > 1000
\quad\Longrightarrow\quad
m > 10.
Since m is a natural number, we have
m \in \{11, 12, 13, \dots\}.
Combining both inequalities from above:
10 < m \le 22.
Hence the integer values for m that satisfy both conditions are:
m \in \{11, 12, 13, \ldots, 22\}.
Step 5: Count the number of possible values for m
The set \{11, 12, \ldots, 22\} has 12 integers (from 11 through 22 inclusive). Therefore, there are
22 - 11 + 1 = 12
possible values of m .
Final Answer
12
