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 n-th term of the sequence, a (n), is given recursively by

• a (1) = 1

• a (n) = a (n - 1) + 21 … … … for n > 1

We can find the explicit rule for the sequence by iterative substitution:

a (2) = a (1) + 21

a (3) = a (2) + 21 = (a (1) + 21) + 21 = a (1) + 2×21

a (4) = a (3) + 21 = (a (1) + 2×21) + 21 = a (1) + 3×21

and so on, with the general pattern

a (n) = a (1) + 21 (n - 1) = 21n - 20

Now, we're told that the sum of some number N of terms in this sequence is 2332. In other words, the N-th partial sum of the sequence is

a (1) + a (2) + a (3) + … + a (N - 1) + a (N) = 2332

or more compactly,

$\displaystyle\sum_{n=1}^N a(n) = 2332$

It's important to note that N must be some positive integer.

Replace a (n) by the explicit rule:

$\displaystyle\sum_{n=1}^N (21n-20) = 2332$

Expand the sum on the left as

$\displaystyle 21 \sum_{n=1}^N n-20\sum_{n=1}^N1 = 2332$

and recall the formulas,

$\displaystyle\sum_{k=1}^n1=\underbrace{1+1+\cdots+1}_{n\text{ times}}=n$

$\displaystyle\sum_{k=1}^nk=1+2+3+\cdots+n=\frac{n(n+1)}2$

So the sum of the first N terms of a (n) is such that

21 × N (N + 1)/2 - 20N = 2332

Solve for N :

21 (N ² + N) - 40N = 4664

21 N ² - 19 N - 4664 = 0

Now for the problem I mentioned at the start: this polynomial has no rational roots, and instead

N = (19 ± √392,137)/42 ≈ -14.45 or 15.36

so there is no positive integer N for which the first N terms of the sum add up to 2332.

4 0

2 years ago

Determine which of the four levels of measurement (nominal, ordinal, interval, ratio) is most appropriate. ages of children: 3

gayaneshka [121]

Answer:

The level of variable ages of children is ratio.

Explanation:

Nominal scale is the one which takes categories as its values like gender of a person.

Ordinal scale is the one which is used to show the order of values with no clear difference among them like satisfaction level of people with a service.

Interval scale is the one which shows the clear order of values with clear difference between the values with out true zero.

Ordered scale is one which not only produces the order of variables but also makes the difference between variables known along with information on the value of true zero.

Therefore, the variable which is under consideration here is the ages of children, which takes the values 3,4,5,6 and 7 is a ratio scale.

6 0

3 years ago

Find the area of the circle. Use 3.14 for it. ( d = 8 cm A = [?] cm2 A=Tr2

storchak [24]

What the make no sense but I believe it is 50.24 hope that helps

8 0

2 years ago

use the method of successive differences to determine the next term in the sequence: 4, 14, 82, 268, 657, 1359

IRISSAK [1]

The method of successive differences uses subtraction to the one number to the next and the process goes on until the difference settles constant which is not equal to zero. In this case, the constant difference reaches 25. Reversing the process to get the next term, the answer is 2509.

6 0

2 years ago