perplexus.info :: Just Math : Sum Greatest And Least Roots

Sum Greatest And Least Roots (Posted on 2008-05-22)

Derive a formula for evaluating the following expression in terms of p and its higher powers, given that p is a positive integer.

   Σ  ([³√m] + <³√m>) 
m=1 to p³

Note: [x] is the greatest integer ≤ x, and <x> is the least integer ≥ x

Submitted by K Sengupta

Rating: 5.0000 (1 votes)

Solution: (Hide)

f(x) = [³√x] + <³√x> (say)

Then, assuming that j is an integer, we must have:

f(x) = 2j-1, whenever (j-1)³ < x ≤ j³-1, and:

f(x) = 2j, whenever x = j³

Also, j³ - 1 – (j-1)³ = 3j(j-1)

Thus,

Σ_{m= 1 to p³} ( [³√m] + <³√m>)

= Σ_{m= 1 to p} (3m(m-1)(2m-1) + 2m)

= (3/2)*p²*(p+1) ² – (3/2)*p(p+1)(2p+1) + (5/2)*p(p+1)

= p(p+1)(3p²- 3p+2)/2 (upon simplification)

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
re: answer	Charlie	2008-05-23 00:10:42
answer	Praneeth	2008-05-22 18:04:48
thoughts	Charlie	2008-05-22 17:23:07

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