If x >17 and (x + 1) (x +2 ) (x+3) (x+4) + c is a perfect square, then find the least non-negative value of c?
@shashankbapat23 said:If x >17 and (x + 1) (x +2 ) (x+3) (x+4) + c is a perfect square, then find the least non-negative value of c?
1?
@shashankbapat23 said:If x >17 and (x + 1) (x +2 ) (x+3) (x+4) + c is a perfect square, then find the least non-negative value of c?
Is it 1?
@shashankbapat23 said:Is goodapproach please
(x + 1) (x +2 ) (x+3) (x+4) + c =a^2
a^-c=(x + 1) (x +2 ) (x+3) (x+4)
=(x+1)(x+5)(x+2)(x+3)
=(x^2+5x+4)(x^2+5x+6)
=(x^+5x)^2+10(x^2+5x)+24
=(x^2+5x+5)^2-1
Comparing both sides
c=1
a^-c=(x + 1) (x +2 ) (x+3) (x+4)
=(x+1)(x+5)(x+2)(x+3)
=(x^2+5x+4)(x^2+5x+6)
=(x^+5x)^2+10(x^2+5x)+24
=(x^2+5x+5)^2-1
Comparing both sides
c=1
Suppose a function is defined over a set of natural numbers as follows:
f(1)=1,f(2)=1,f(3)=-1,and f(n)=f(n-1)f(n-3) whenever n>3.Find the value of
f(694)+f(695) ?
f(1)=1,f(2)=1,f(3)=-1,and f(n)=f(n-1)f(n-3) whenever n>3.Find the value of
f(694)+f(695) ?
@shashankbapat23 said:Suppose a function is defined over a set of natural numbers as follows:f(1)=1,f(2)=1,f(3)=-1,and f(n)=f(n-1)f(n-3) whenever n>3.Find the value of f(694)+f(695) ?
2
@shashankbapat23 said:correct approach please
7 ki cycle hai which goes like this
1,1,-1,-1,-1,1,-1
Then it repeats
f(694)=>694 mod 7=1
and 695 mod 7=2
First 2 terms=1+1=2
1,1,-1,-1,-1,1,-1
Then it repeats
f(694)=>694 mod 7=1
and 695 mod 7=2
First 2 terms=1+1=2
@shashankbapat23 said:Suppose a function is defined over a set of natural numbers as follows:f(1)=1,f(2)=1,f(3)=-1,and f(n)=f(n-1)f(n-3) whenever n>3.Find the value of f(694)+f(695) ?
ans : 2
values start repeating after 7 terms hence cyc. is 7
which gen. values as:
1,1,-1,-1,-1,1,-1
f(694/7)=f(1)=1
f(695/7)=f(2) i.e. 2nd term of series=1
so f(694)+f(695)=1+1=2
values start repeating after 7 terms hence cyc. is 7
which gen. values as:
1,1,-1,-1,-1,1,-1
f(694/7)=f(1)=1
f(695/7)=f(2) i.e. 2nd term of series=1
so f(694)+f(695)=1+1=2
For all real x, f(x) satisfies 2f(x) + f(1-x) = 2*x2 + 1. Then which among the following is true for all x?
a) f(x) b) f(x) >= -1
c) f(x) >= 1
d) f(x)
c) f(x) >= 1
d) f(x)
@shashankbapat23 said:For all real x, f(x) satisfies 2f(x) + f(1-x) = 2*x2 + 1. Then which among the following is true for all x?a) f(x) b) f(x) >= -1c) f(x) >= 1d) f(x)
(b)?
regards
scrabbler
regards
scrabbler
@shashankbapat23 said:For all real x, f(x) satisfies 2f(x) + f(1-x) = 2*x2 + 1. Then which among the following is true for all x?a) f(x) b) f(x) >= -1c) f(x) >= 1d) f(x)
solving above we will get f(x)=(2/3)x^2+(4/3)x-1/3
this quadratic equation will give minimum value at x=-1,and min value f(x)=-1
hence option b
this quadratic equation will give minimum value at x=-1,and min value f(x)=-1
hence option b
@shashankbapat23 said:For all real x, f(x) satisfies 2f(x) + f(1-x) = 2*x2 + 1. Then which among the following is true for all x?a) f(x) b) f(x) >= -1c) f(x) >= 1d) f(x)
2f(x) + f(1 - x) = 2x² + 1 ..................(1)
put x = 1 - y
2f(1 - y) + f(y) = 2(1 - y)² + 1
so, 2f(1 - x) + f(x) = 2x² - 4x + 3 ...............(2)
Subtract (2) from twice of (1) to get get
3f(x) = 2x² + 4x - 1 = 2(x + 1)² - 3
f(x) = 2(x + 1)²/3 - 1
So, f(x) >= -1
@shashankbapat23 said:Is good.please share your approach
Have some patience yaar, someone will definitely help you with the appraoch
@shashankbapat23 said:For all real x, f(x) satisfies 2f(x) + f(1-x) = 2*x2 + 1. Then which among the following is true for all x?a) f(x) b) f(x) >= -1c) f(x) >= 1d) f(x)
@shashankbapat23 said:Is good.please share your approach
Sorry comp rebooted while I was typing it out :(
Rather than "solve" formally, I tried to find a couple of values.
Let f(1) = a and f(0) = b. Then 2a + b = 3 and 2b + a =1 which gives a = 5/3 and b = -1/3. The only answer range which contained both was (b) hence....
regards
scrabbler
Rather than "solve" formally, I tried to find a couple of values.
Let f(1) = a and f(0) = b. Then 2a + b = 3 and 2b + a =1 which gives a = 5/3 and b = -1/3. The only answer range which contained both was (b) hence....
regards
scrabbler
A, B and C enter into a partnership and invest their amounts in the ratio 7: 4 : 6 respectively. After 4 months, A increases his share by 50%. If the total profit at the end of a year is Rs. 87,000, then find B €™s share in the profit.
20k
18k
17k
16k
20k
18k
17k
16k
@Cat.Aspirant123 said:A, B and C enter into a partnership and invest their amounts in the ratio 7: 4 : 6 respectively. After 4 months, A increases his share by 50%. If the total profit at the end of a year is Rs. 87,000, then find B €™s share in the profit.20k18k17k16k
18k