Explanation
Solution
Step 1: Series ka Pattern pehchanna
Is series ka har bracket ek Geometric Progression (G.P.) ka sum hai jahan terms alternating positive aur negative hain.
Hum jaante hain ki:
an−bn=(a−b)(an−1+an−2b+⋯+bn−1)
Isi tarah, agar hum n-th bracket ko dekhen:
Tn=2n1−2n−1⋅31+2n−2⋅321−⋯+(−1)n3n1
Yeh ek G.P. hai jisme first term a=2n1 hai, common ratio r=−32 hai, aur total (n+1) terms hain.
G.P. sum formula (S=1−ra(1−rk)) ka use karke:
Tn=1−(−32)2n1[1−(−32)n+1]=352n1[1−(−32)n+1]
Tn=53[2n1−2n1⋅3n+1(−2)n+1]
Tn=53[2n1−3n+1(−1)n+1⋅2]
Step 2: Total Sum (S) nikalna
Poori series ka sum S=∑n=1∞Tn hoga:
S=53[n=1∑∞(21)n−2n=1∑∞3n+1(−1)n+1]
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Pehla part: ∑n=1∞(21)n=1−1/21/2=1
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Dusra part: ∑n=1∞3n+1(−1)n+1
Iske terms honge: 321−331+341…
Yeh ek infinite G.P. hai (a=1/9,r=−1/3):
Sum=1−(−1/3)1/9=4/31/9=121
Step 3: Value put karna
S=53[1−61]=53⋅65=63=21
Yahan βα=21.
Kyunki 1 aur 2 co-prime hain, isliye:
Step 4: Final Calculation
Hamein α+3β nikalna hai:
Answer:
α+3β ki value 7 hai.
