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Question

If α1,α2,α3,......,αn are n-values of a variable α such that ni=1ui=155 and ni=1vi=225, where ui=2αi3 and vi=3αi5, for i = 1, 2, 3, ..., n, then the value of (ni=1ui+ni=1vin) is

A
25.¯3
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B
35.¯3
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C
27.¯9
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D
25
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Solution

The correct option is A 25.¯3
Given: ni=1ui=155 and ni=1vi=225
where ui=2αi3,vi=3αi5; i = 1, 2, ..., n,
Now, ni=1ui+ni=1(2αi3)
155=2ni=1αi3n
155=2(α1+α2+....+αn)3n
α1+α2+....+αn=155+3n2............(i)
Similarly, ni=1vi=ni=1(3αi5)
225=3ni=1αi5n
225=3(α1+α2+....+αn)5n
α1+α2+....+αn=225+5n3............(ii)
Equating (i) and (ii), we get
155+3n2=225+5n3
465+9n=450+10n
15=n
Putting all the values in (ni=1ui+ni=1vin), we get
ni=1ui+ni=1vin=155+22515
=38015
=763
= 25.33....
=25.¯3
Hence, the correct answer is option (a).

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