For any odd integer n ≥ 1, n3 – (n-1)3 + ……….+ (-1)n-1 13 =…………
Shane Macguire , 10 Years ago
Grade upto college level
Algebra> For any odd integer n ≥ 1, n 3 – (n-1) 3 ...
1 AnswersDeepak Patra
Last Activity: 10 Years ago
Hello Student,
Please find the answer to your question
Ans. Since n is an odd integer, (-1)n-1 = 1 and n – 1, n – 3, n – 5, … are even integers.
We have
n3 + (n – 1)3 + (n – 2) 3 – (n – 3) 3 + …. + (-1)n – 1 13
= n3 + (n – 1) 3 + ( n – 2) 3 + ….. + 13 – 2[(n - 1) 3 + (n – 3) 3 + …… 23 ]
=[n3 + (n – 1)3 +( n – 2) 3 +……. + 13]
-2 * 23 [(n – 1/2) 3 + (n – 3/2) 3 + ….. + 13 ]
[ ∵ n – 1, n – 3 …………………….. are even integers]
Here the first square bracket contain the sum of cubes of Ist n natural numbers. Whereas the second square bracket contains the sum of the cubes of natural numbers from l to
(n – 1 /2), where n – 1, n – 3, ……………… are even integers. Using the formula for sum of cubes of lst n natural numbers we get the summation
= [ n (n + 1)/2]2 – 16 [ { 1/2 ( n – 1/2)( n – 1/2 + 1)]2
= 1/4n2 (n +1) 2 – 16 (n – 1) 2 ( n + 1) 2 /16 * 4
= 1/4 ( n + 1) 2 [n2 - ( n - 1) 2 = 1/4 ( n + 1) 2 (2n – 1)
Thanks
Deepak Patra
askIITians Faculty
Provide a better Answer & Earn Cool Goodies
Enter text here...
LIVE ONLINE CLASSES
Prepraring for the competition made easy just by live online class.
Full Live Access
Study Material
Live Doubts Solving
Daily Class Assignments
Ask a Doubt
Get your questions answered by the expert for free
Enter text here...
Other Related Questions on algebra
Can we divide two vectors? If this is possible then how?
algebra
1 Answer Available
Last Activity: 1 Year ago
How do you write “the product of 18 and q” as an algebraic expression?
algebra
1 Answer Available
Last Activity: 1 Year ago
दहावीच्या आय सी एस इ विद्यार्थ्यांनी पुढे काय करायला पाहिजे
algebra
0 Answer Available
Last Activity: 2 Years ago