Let a, b, c > 1, a^3, b^3 and c3 be in A.P., and logab,logca and logbc be in G.P. If the sum of first 20 terms of an A.P., whose first term is 3a+4b+c and the common difference is 10a−8b+c is −444, then abc is equal to:
Explanation
Calculation
1. G.P. Condition Se:
Chunki logab,logca,logbc G.P. mein hain:
Maan lijiye logca=x, toh logac=x1:
Iska matlab logca=1⟹a=c.
2. A.P. Condition Se:
Chunki a3,b3,c3 A.P. mein hain aur a=c:
Iska matlab a=b=c.
3. Sum of A.P. Use Karte Hue:
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First term A=3a+4a+a=36a=2a
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Common difference D=10a−8a+a=10−6a=−0.6a
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Sum S20=220[2A+(20−1)D]=−444
4. Final Value:
Chunki a=b=c=6:
Sahi Option: (2)
Explanation
Calculation
1. G.P. Condition Se:
Chunki logab,logca,logbc G.P. mein hain:
Maan lijiye logca=x, toh logac=x1:
Iska matlab logca=1⟹a=c.
2. A.P. Condition Se:
Chunki a3,b3,c3 A.P. mein hain aur a=c:
Iska matlab a=b=c.
3. Sum of A.P. Use Karte Hue:
-
First term A=3a+4a+a=36a=2a
-
Common difference D=10a−8a+a=10−6a=−0.6a
-
Sum S20=220[2A+(20−1)D]=−444
4. Final Value:
Chunki a=b=c=6:
Sahi Option: (2)
