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Show that 11 2 + 12 2 +13 2 + .............+ 20 2 is an odd integer divisible by 5.

Show that  11 2 + 12 2 +13 2 + .............+ 20 2 is an odd integer divisible by 5.

Grade:12

2 Answers

AskIITians Expert Hari Shankar IITD
17 Points
12 years ago

Sum of squares of first n consecutive integers = n(n+1)(2n+1)/6.

Therefore, 12 +22 + ...+192 +202 = 20 * 21 * 41 /6 = 2870.............(1)

Also, 12 + 22 + 32 + ...+ 102 = 10*11*21/6 = 385...................(2)

 

Subtracting (2) from (1),

the first 10 terms of (1) get cancelled out and we get

112 + 122 + 132 = ....+ 192 +202 = 2870-385 = 2485, which is an odd integer and is divisible by 5

 

 

Ramesh V
70 Points
12 years ago

This is simply:

S= sum ( 12+22+32+...202) - sum (12+22+32+...102)

as (12+22+32+........+n2 ) = n(n+1)(2n+1)/6

S = 2485 which is divisible by 5..

If this answer is wat u r not expecting, plzz post question properly ..

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