Is this simply the expansion of (a-b)^2?
5 5 5 5 4 2 − ( 2 ) 5 5 5 5 5 2 + 5 5 5 5 6 2 = ?
The answer is 2.
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12 solutions
Moderator note:
Using the same approach, can you evaluate 5 5 5 5 4 3 − ( 3 ) 5 5 5 5 5 3 + ( 3 ) 5 5 5 5 6 3 − 5 5 5 5 7 3 ?. Can you spot a pattern here?
Great Nihar... The best solution. I used the same method....
same method!nice prob!
but there was no need to use a 2 − b 2 ,you could just have used the simple formula, ( a + b ) 2 + ( a − b ) 2 = 2 ( a 2 + b 2 ) !! That is what I used!
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Well, both methods are simple.
Well , I want your deathnote on rent. :P
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hahaha!! but then how will i collect the rent!!!
This is a key to a general solution
Let x be a real number, and n a natural number, then it can be shown that:
k = 0 ∑ n ( k n ) ⋅ ( − 1 ) k ⋅ ( x + k ) n
is equal to
k = 0 ∑ n ( k n ) ⋅ ( − 1 ) k ⋅ k n
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And how would you prove that?
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Sir, I noticed that the coefficients when arranged form the pascals triangle.
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@Nihar Mahajan – Can you elaborate on it?
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@Brilliant Mathematics – Consider that the signs are removed.If degree of expression is 2 , then coefficients are of the 2 n d row of pascal's triangle.If degree of expression is 3 , then coefficients are of the 3 r d row of pascal's triangle.So , If degree of expression is n , then coefficients are of the n t h row of pascal's triangle.So , the general form must include binomial coefficients.
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@Nihar Mahajan – Yes, that's what Gustavo Merchan is trying to convey. See my note in his solution below.
I'll try to formalize it a bit and post it
I did it a little differently. Instead of using difference of square formula, I expanded the expression -
= ( a − 1 ) 2 − 2 a 2 + ( a + 1 ) 2 = a 2 − 2 a + 1 − 2 a 2 + a 2 + 2 a + 1 = 2
Moderator note:
Yes, this is the shortest solution to obtain the answer. Good job!
same method by me
Challenge Student's note: Thanks a lot!
suppose there are three consecutive numbers a,b,c Now we know that for any three consecutive numbers a,b,c b^2=a*c+1 So A^2-2B^2+C^2= A^2-2(AC)-2+C^2 =(C-A)^2 -2 =2^2 -2 =4-2 =2
55554^2 = 3086246916 55555^2 = 3086358025 55556^2 = 3086469136
3086246916 - 2(3086358025) + 3086469136 = 2
Let 55555 =a Equation becums
(a-1)^2 - 2a^2.+ (a+1)^2
a^2 +1 - 2a - 2a^2 +a^2 +1 +2a = 2a^2 - 2a^2 +2a-2a +1+1 =0 +0 +2 =2
SIMPLY TAKE 55555 = X and u get the answer
AND
ANSWER WILL BE DEFINETLY 2
Let 55555= x =(x-1)^2 - 2x^2 + (x+1)^2 =(x-1)^2 -x^2 +(x+1)^2 -x^2 =(x-1+x)(x-1-x) + (x+1+x)(x+1-x) =(2x-1)(-1)+(2x+1)(1) =-2x + 1 + 2x+ 1 =2
Let 5 5 5 5 6 = b and 5 5 5 5 4 = a . The expression will be:
= a 2 − 2 [ 2 ( a + b ) ] 2 + b 2 = a 2 − 2 ( a 2 + 2 a b + b 2 ) + b 2 = 2 1 ( a 2 − 2 a b + b 2 ) = 2 1 ( a − b ) 2 = 2 1 ( 5 5 5 5 6 − 5 5 5 5 4 ) 2 = 2 2 2 = 2
This is a key to a general solution
Let x be a real number, and n a natural number, then it can be shown that:
k = 0 ∑ n ( k n ) ⋅ ( − 1 ) k ⋅ ( x + k ) n = k = 0 ∑ n ( k n ) ⋅ ( − 1 ) k ⋅ k n
for n=2, result is 2, independently of x
for n=3, result is -6, independently of x
for n=4, result is 24, independently of x
for n=5, result is -120, independently of x
etc.
Moderator note:
Does the numbers 2 , 6 , 2 4 , 1 2 0 look familiar to you? Is there a general formula for all n independent of x ?
oohh !! it's
( − 1 ) n ⋅ n !
I hadn't noticed that ... thanks :)
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Can you prove that it's true for all n ?
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Let me try it a bit more ..., I have a new idea
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@Gustavo Merchan – This is 6 months old but can anyone prove the general case is (-1)^n*n! ? I'd be curious to see the proof.
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@Jonathan Hocker – after pages and pages of drafting, couldn't get past a certain point in the proof... But given your interest, I will get back to it soon
Let, 55554 =x now, the expression is, x^2 -2(x+1)^2+(x+2)^2 =2.
[(55554)^2 -(55555)^2]+ [(55556)^2-(55555)^2 ]= C]= [(55556)-(55555)]+[(55556)+(55555)] +[(55554) -(55555)] +[(55554) +(55555)] = 111111 * 1-111109 * -1 =2.
let 55556= a and 55554= b; (a-b)^2 = a^2 + b^2 - 2
a
b
now
2
a
b= 2* 55556 * 55554= 2(55555+1)(55555-1)=2
(55555)^2 - 2
(a-b)^2 = 2^2 = 4
using these values
4= a^2 + b^2 - 2
(55555)^2+2
=> 4-2= (55556)^2 + (55554)^2 - 2
(55555)^2
=> (55556)^2 + (55554)^2 - 2
(55555)^2= 2 [ANS]
Let 5 5 5 5 5 = a , which makes the given expression :
( a − 1 ) 2 − 2 a 2 + ( a + 1 ) 2 = ( a − 1 ) 2 − a 2 + ( a + 1 ) 2 − a 2 = ( a − 1 + a ) ( a − 1 − a ) + ( a + 1 − a ) ( a + 1 + a ) = ( 2 a − 1 ) ( − 1 ) + ( 2 a + 1 ) ( 1 ) = − 2 a + 1 + 2 a + 1 = 2