Given equation My solution:- (40×41×42×43) + 1 = (40×43)×(41×42) + 1 = (1720)×(1722) + 1 = (1721-1)×(1721+1) + 1 = 1721^2 - 1^2 + 1 = 1721^2 So, √((40×41×42×43)+1) = √(1721^2) = 1721 It is more fast and efficient method But,your method is very good sir 👍
Consider the following: x(x+1)(x+2)(x+3)=(x²+3x)(x²+3x+2) =[(x²+3x+1)-1][(x²+3x+1)+1] =(x²+3x+1)²-1 With x=40 then 40×41×42×43+1=(40²+3×40+1)² =(1600+120+1)² =1721² Thus sqrt[(40×41×42×43)+1]=1721
@@SALogics Go a step further and note both n^2 + 3n + 1 = (n+0) (n+3) + 1 = (n+1) (n+2) - 1 allowing one to (even more) quickly calculate the answer in a few seconds with mental math.
40*41*42*43 + 1 = x^2 x^2 - 1 = 40*41*42*43 (x-1)(x+1) = 40*41*42*43 Observe that (x+1) - (x-1) = 2 We know x-1 is not 40 * 42 because 40 * 42 ends in zero and 41 * 43 ends in 3 a difference of 3 or more. Whereas x-1 = 40*43 ends in 0 and x+1 = 41*43 ends in 2. So a difference of 2 is possible. 40*43 = 1720 and 41*42 = 1722. Thus x = 1721 and that is the answer.
Given equation
My solution:-
(40×41×42×43) + 1
= (40×43)×(41×42) + 1
= (1720)×(1722) + 1
= (1721-1)×(1721+1) + 1
= 1721^2 - 1^2 + 1
= 1721^2
So,
√((40×41×42×43)+1) = √(1721^2)
= 1721
It is more fast and efficient method
But,your method is very good sir 👍
Very nice! ❤
Consider the following:
x(x+1)(x+2)(x+3)=(x²+3x)(x²+3x+2)
=[(x²+3x+1)-1][(x²+3x+1)+1]
=(x²+3x+1)²-1
With x=40 then
40×41×42×43+1=(40²+3×40+1)²
=(1600+120+1)²
=1721²
Thus sqrt[(40×41×42×43)+1]=1721
Very nice trick! ❤
👍👍👏👏👏
Thanks for liking! ❤
Nice
Very nice! ❤
Excellent
Thank you so much 😀
More elegant x=m^2+3m+1. =1600+120+1=1721. Sqr((x-1)*(x+1)-1)=±x=±1721
Very nice! ❤
There are around 500 versions of this problem at 500 different channels
Sqrt(n(n+1)(n+2)(n+3) + 1) = n^2 + 3n + 1
Very nice! ❤
@@SALogics Go a step further and note both
n^2 + 3n + 1 = (n+0) (n+3) + 1
= (n+1) (n+2) - 1
allowing one to (even more) quickly calculate the answer in a few seconds with mental math.
Shortcuts for this form are:
1. 40(43) + 1 = 1721;
OR
2. 41(42) - 1 = 1721.
Either n(n + 3) + 1, or
(n + 1)(n + 2) - 1.
Very nice! ❤
Terrific!🎉😂
40*41*42*43 + 1 = x^2
x^2 - 1 = 40*41*42*43
(x-1)(x+1) = 40*41*42*43
Observe that (x+1) - (x-1) = 2
We know x-1 is not 40 * 42 because 40 * 42 ends in zero and 41 * 43 ends in 3 a difference of 3 or more.
Whereas x-1 = 40*43 ends in 0 and x+1 = 41*43 ends in 2. So a difference of 2 is possible.
40*43 = 1720 and 41*42 = 1722. Thus x = 1721 and that is the answer.
Very nice trick! I really appreciate that ❤
Neat
Thanks and welcome! ❤
Wonderful Sir 🙏🙏🙏
Thank a lot! ❤
La gracia es aplicar artificios álgebraicos de manera que al final podamos reemplazar y el cálculo aritmético sea sea sencillo... muy bien explicado
¡muchas gracias! ❤
The product of four consecutive integer numbers plus 1 is always a perfect square.
Very nice! ❤
X=sqr of (40.41.42.43+1)
X^2-1=40.43.41.42=
1720.1722=(x-1)(x+1)
X-1=1720 x=1721
Very nice trick! ❤
Решил устно. Конечно,знал метод.
Да, ❤
Beautiful ❤🎉
Thank you! 😊 ❤
?=40^2+3.40+1=1600+121=16121.ans
Very nice! ❤
let x=m^2+3m+1, much easier.
You are right! ❤
167
The answer is 1721 ❤
how about
√(40•41•42•43 + 1) =
√(80•82•84•86 + 16) / 4 =
√((83 - 1)•(83 + 1)•(83 - 3)•(83 + 3) +16) / 4 =
√((83² - 1)•(83² - 9) +16) / 4 =
√(83⁴ - 10•83² +25) / 4 =
√((83² - 5)² / 4 =
(83² - 5) / 4 = 1721
Very nice trick! ❤
1721
Yes, you are right! ❤
@@SALogics -1,721 is another solution
? = 1721
You are right! ❤
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