Thirty two divided by the opposite of 4
2 answers:
You might be interested in
There's a slight problem with your question, but we'll get to that...
Consecutive terms of the sequence are separated by a fixed difference of 21 (22 = 1 + 21, 43 = 22 + 21, 64 = 43 + 21, and so on), so the <em>n</em>-th term of the sequence, <em>a</em> (<em>n</em>), is given recursively by
• <em>a</em> (1) = 1
• <em>a</em> (<em>n</em>) = <em>a</em> (<em>n</em> - 1) + 21 … … … for <em>n</em> > 1
We can find the explicit rule for the sequence by iterative substitution:
<em>a</em> (2) = <em>a</em> (1) + 21
<em>a</em> (3) = <em>a</em> (2) + 21 = (<em>a</em> (1) + 21) + 21 = <em>a</em> (1) + 2×21
<em>a</em> (4) = <em>a</em> (3) + 21 = (<em>a</em> (1) + 2×21) + 21 = <em>a</em> (1) + 3×21
and so on, with the general pattern
<em>a</em> (<em>n</em>) = <em>a</em> (1) + 21 (<em>n</em> - 1) = 21<em>n</em> - 20
Now, we're told that the sum of some number <em>N</em> of terms in this sequence is 2332. In other words, the <em>N</em>-th partial sum of the sequence is
<em>a</em> (1) + <em>a</em> (2) + <em>a</em> (3) + … + <em>a</em> (<em>N</em> - 1) + <em>a</em> (<em>N</em>) = 2332
or more compactly,
It's important to note that <em>N</em> must be some positive integer.
Replace <em>a</em> (<em>n</em>) by the explicit rule:
Expand the sum on the left as
and recall the formulas,
So the sum of the first <em>N</em> terms of <em>a</em> (<em>n</em>) is such that
21 × <em>N</em> (<em>N</em> + 1)/2 - 20<em>N</em> = 2332
Solve for <em>N</em> :
21 (<em>N</em> ² + <em>N</em>) - 40<em>N</em> = 4664
21 <em>N</em> ² - 19 <em>N</em> - 4664 = 0
Now for the problem I mentioned at the start: this polynomial has no rational roots, and instead
<em>N</em> = (19 ± √392,137)/42 ≈ -14.45 or 15.36
so there is no positive integer <em>N</em> for which the first <em>N</em> terms of the sum add up to 2332.
James'age : 12, Kate's age : 7
<h3>Further Explanation </h3>One variable linear equation is an equation that has a variable and the exponent number is one. Can be stated in the form:
ax + b = c, where a, b, and c are constants, x is a variable
Whereas the two-variable linear equation is a linear equation that has 2 variables and the exponent is one
Can be stated in the form:
x, y = variable
In the statement of the task:
- James is two years younger than twice his sister Kate’s Age can be stated in the equation: James (J) = 2K (Kate) - 2 ⇒equation 1
- Together their ages add up to 19. The equation: J + K = 19 .... 2 ---> J = 19 - K ⇒equation 2
So the solution is:
Equation 2 is entered into equation 1 so that it becomes:
19-K = 2K-2
21 = 3K
K = 7
because J + K = 19, then the value of J = 19-7 = 12
<h3>Learn more</h3>the slope-intercept form of the linear equation
a graph of a linear equation
system of linear equations
Keywords : Age, variable, linear equation
