eleven =839 twelve =1547 thirteen =?

Question

ScanSolve AI Tutor · Accepted Answer

To find the pattern, let's represent "eleven" as $n=11$ and "twelve" as $n=12$ . We are looking for a function $f(n)$ such that $f(11) = 839$ and $f(12) = 1547$ .

Let's assume the pattern is a polynomial. Given two points, we can uniquely determine a linear function. Given three points, a quadratic. Given four points, a cubic. Since we only have two points, a general cubic or higher-order polynomial is not uniquely determined. However, these types of problems often imply a specific form of polynomial or a simple arithmetic pattern.

Let's try a pattern of the form $f(n) = n^3 + An + B$ . This form is a cubic polynomial where the $n^2$ and constant terms are related in a specific way, allowing it it to be determined by two points.

Formulate the pattern
Let the number be $n$ . We hypothesize a pattern of the form $f(n) = n^3 + An + B$ .
For "eleven", $n=11$ , $f(11) = 839$ .
For "twelve", $n=12$ , $f(12) = 1547$ .
Set up equations for A and B
Substitute the given values into the hypothesized function:
For $n=11$ :
$11^3 + 11A + B = 839$
$1331 + 11A + B = 839$
$11A + B = 839 - 1331$
$11A + B = -492 \quad (1)$

For $n=12$ :
$12^3 + 12A + B = 1547$
$1728 + 12A + B = 1547$
$12A + B = 1547 - 1728$
$12A + B = -181 \quad (2)$
Solve for A
Subtract equation (1) from equation (2):
$(12A + B) - (11A + B) = -181 - (-492)$
$A = -181 + 492$
$A = 311$
Solve for B
Substitute $A=311$ into equation (1):
$11(311) + B = -492$
$3421 + B = -492$
$B = -492 - 3421$
$B = -3913$
Determine the function
The pattern function is $f(n) = n^3 + 311n - 3913$ .
Calculate the value for "thirteen"
For "thirteen", $n=13$ :
$f(13) = 13^3 + 311(13) - 3913$
$f(13) = 2197 + 4043 - 3913$
$f(13) = 6240 - 3913$
$f(13) = 2327$

The value for "thirteen" is:
2327

Eleven =839 twelve =1547 thirteen =?

Answer

Step-by-step solution

Formulate the pattern

Set up equations for A and B

Solve for A

Solve for B

Determine the function

Calculate the value for "thirteen"

Try one yourself

Still stuck on this one?

More Mathematics Questions