What is the maximum value of $d_n$ and for which value of $n$ does it occur?

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In summary, the maximum value of $d_n$ is 401, which is achieved when $n = 401k + 200$ for some integer $k$.
  • #1
Albert1
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$ a_n=100+n^2.\,\, n=1,2,3,----$

$d_n=(a_n,a_{n+1})$

find :max($d_n)$
 
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  • #2
Albert said:
$ a_n=100+n^2.\,\, n=1,2,3,----$

$d_n=(a_n,a_{n+1})$

find :max($d_n)$
[sp]If $d$ divides $a_n$ and $a_{n+1}$ then $d$ divides $a_{n+1} - a_n = 2n+1$. So $d$ divides $2a_n - n(2n+1) = 200-n$. Therefore $d$ divides $(2n+1) + 2(200-n) = 401$. But $401$ divides $a_{200}$ ($= 40100$) and $a_{201}$ ($= 40501$). Thus $\max d_n = 401.$[/sp]
 
  • #3
dn= (100 + n^2, 100 + (n+1)^2)
= (100+n^2 , 100 + n^2 + 2n + 1)
= (100+n^2, 2n + 1) subtracting 1st term from second
= ( 200+2n^2 , 2n + 1) doubling 1st as 2nd is odd
= (200+ 2n^2- n(2n+1), 2n+ 1)
= ( 200 - n , 2n + 1)
= (400- 2n ,2n + 1) doubling 1st as second is odd
= ( 400 -2 n + 2n+ 1,2n+1)
= (401,2n + 1)
it shows
1) cannot be > 401
2) is 401 when 2n +1 = odd multiple of 400
or 2n + 1 = 401(2k+ 1) = 802k + 401

or 2n = 802k + 400

or n = 401 k + 200
we are able to find n as well for which it is 401
 

FAQ: What is the maximum value of $d_n$ and for which value of $n$ does it occur?

What does the $d_n$ notation mean in "Max of $d_n$: $100 + n^2$"?

The $d_n$ notation represents the domain of the function, or the input values that the function can take. In this case, the function is defined as $100 + n^2$, where n represents the input value.

How do you find the maximum value of the function $100 + n^2$?

To find the maximum value of the function, we can take the derivative of the function and set it equal to 0. This will give us the critical points of the function, and we can then determine which point corresponds to the maximum value. In this case, the maximum value will occur at n=0, with a value of 100.

Can you explain the significance of the function $100 + n^2$ in scientific research?

The function $100 + n^2$ is a simple quadratic function that can be used to model a variety of real-world phenomena. It can be used to represent anything from the growth of a population to the trajectory of a projectile. In scientific research, this function can be used as a starting point for more complex models or to analyze data and make predictions.

How would the maximum value of the function change if the value of n is increased?

As n increases, the maximum value of the function will also increase. This is because the function $n^2$ is an increasing function, meaning that as the input value increases, the output value also increases. Therefore, as n increases, the value of $100 + n^2$ will also increase, resulting in a higher maximum value.

Can the maximum value of the function ever be negative?

No, the maximum value of the function $100 + n^2$ will never be negative. This is because the function has a minimum value of 100, which occurs at n=0. As n increases, the value of the function will only increase, never reaching a negative value. Therefore, the maximum value of this function will always be a positive number.

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