Explanation
Step 1: G.P. ki terms ko define karna
Maana ki G.P. ki pehli term a hai aur common ratio r hai. Kyunki G.P. "increasing positive numbers" ki hai, isliye a > 0 aur r > 1.
Step 2: Di gayi conditions se equations banana
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Product of 3rd and 5th terms:
a3⋅a5=91
(ar2)⋅(ar4)=91
a2r6=91
Dono taraf square root lene par (kyunki numbers positive hain):
ar3=31 --- (Equation 1)
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Sum of 6th and 8th terms:
a6+a8=2
ar5+ar7=2
Isme se r2 common lene par:
r2(ar3+ar5)=2 ya fir ar5(1+r2)=2
Humein pata hai ar5=(ar3)⋅r2. Equation 1 se value rakhne par:
31⋅r2⋅(1+r2)=2
r2(1+r2)=6
r4+r2−6=0
Step 3: r2 ki value nikalna
Quadratic equation solve karte hain (r2 ko x maan kar):
x2+x−6=0
(x+3)(x−2)=0
x=−3 ya x=2
Kyunki r2 negative nahi ho sakta, isliye r2=2.
Step 4: Final expression ki value nikalna
Humein value chahiye: 6(a2+a4)(a4+a6)
6(ar+ar3)(ar3+ar5)
Common nikalne par:
6⋅[ar(1+r2)]⋅[ar3(1+r2)]
6⋅(a2r4)⋅(1+r2)2
Ise hum aise likh sakte hain:
6⋅r2(a2r6)⋅(1+r2)2
Ab values put karte hain (a2r6=91 aur r2=2):
=6⋅21/9⋅(1+2)2
=6⋅181⋅(3)2
=186⋅9
=31⋅9=3
Correct Option: (2) 3
