Explanation
Step 1: Term aur Sum ka relation samjhein
Hume pata hai ki Sn−Sn−1=an (nth term).
Isliye:
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S_6 > S_5 + 1 \implies S_6 - S_5 > 1 \implies a_6 > 1
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S_7 < S_6 + \frac{1}{2} \implies S_7 - S_6 < \frac{1}{2} \implies a_7 < \frac{1}{2}
Step 2: a6 aur a7 ko a4 ki terms mein likhein
Diya gaya hai a4=500 aur common ratio r=m1.
Step 3: Inequalities ko solve karein
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Pehli condition (a_6 > 1):
\frac{500}{m^2} > 1 \implies m^2 < 500
Hum jante hain ki 222=484 aur 232=529.
Isliye, m≤22.
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Doosri condition (a_7 < \frac{1}{2}):
\frac{500}{m^3} < \frac{1}{2} \implies m^3 > 1000
Hum jante hain ki 103=1000.
Isliye, m > 10 (yaani m≥11).
Step 4: Possible values of m nikalna
Dono conditions ko milane par:
Kyunki m ek natural number hai, toh possible values hain:
{11,12,13,14,15,16,17,18,19,20,21,22}
Step 5: Count karein
Total values = 22−11+1=12.
Final Answer:
Possible values of m ki sankhya 12 hai.
