sequences...find it

Algebra Level 4

T ( n ) T(n) refers to the nth term of an arithmetic progression with common difference d d , satisfying T ( 7 ) = 9 T(7) = 9 .

Over all such sequences, what is the value of d d that would minimize

T ( 1 ) × T ( 2 ) × T ( 7 ) ? T(1) \times T(2) \times T(7)?

NONE OF THE ABOVE 33/20 5/4 33/2

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1 solution

Sandeep Bhardwaj
Aug 27, 2014

Let the first term of the A.P. = a

and the common difference = d

so T(7)= a + 6d = 9

=> a = 9 6 d a= 9 - 6d

T ( 1 ) × T ( 2 ) × T ( 7 ) = a ( a + d ) . 9 T(1) \times T(2) \times T(7)= a (a + d).9 ...........................(1)

Putting a = 9 6 d a= 9 - 6d in the equation ( 1 ) (1)

T ( 1 ) × T ( 2 ) × T ( 7 ) = ( 9 6 d ) ( 9 6 d + d ) . 9 T(1) \times T(2) \times T(7)= (9 - 6d) (9 - 6d +d ) .9

Now T ( 1 ) × T ( 2 ) × T ( 7 ) T(1) \times T(2) \times T(7) is a function of d ..

.so let T ( 1 ) × T ( 2 ) × T ( 7 ) = f ( d ) T(1) \times T(2) \times T(7)=f(d) [ f(d) means a function of d ]

so f ( d ) = 9. ( 9 6 d ) ( 9 5 d ) f(d) = 9.(9-6d)(9-5d)

So for getting the least value of f(d), lets differentiate it..and then equate to zero , from there we will get the value of d at which f(d) will be minimum.

f ( d ) = 9 ( 9 6 d ) ( 5 ) + 9 ( 9 5 d ) ( 6 ) f'(d) = 9(9-6d)(-5) + 9(9-5d)(-6)

putting f ( d ) = 0 f'(d)=0 , we get d = 33 20 d=\frac{33}{20}

and also f ( d ) > 0 f''(d) > 0 , so at d = 33 20 d=\frac{33}{20} . f(d) will attain minimum value.

Hence d = 33 20 \boxed{d=\frac{33}{20}}

Thanks for clearing that up. I've edited your question, can you check it to ensure it is correct?

Calvin Lin Staff - 6 years, 9 months ago

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Yes, it's correct sir. Thanks very much.

Sandeep Bhardwaj - 6 years, 9 months ago

Actually i am new to brilliant , so don't have much idea about posting, editing, writing solutions , sets and so on..I think time will heal all these my drawbacks. Thanks again for supporting sir, @Calvin Lin

Sandeep Bhardwaj - 6 years, 9 months ago

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No worries, you're doing great. Practice makes perfect!

Calvin Lin Staff - 6 years, 9 months ago

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Sir, i want to joind beta square, but don't why my visa card ( for payment) is being shown invalid. ???

Sandeep Bhardwaj - 6 years, 9 months ago

You're excellent :D

K.S Anirudh - 6 years, 9 months ago

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