I have two quarters and a dime. How much more money is needed to make $1.00?

How many terms of the arithmetic sequence {1,22,43,64,85,…} will give a sum of 2332? Show all steps including the formulas used

MA_775_DIABLO [31]

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 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

3 years ago

Find the area of polygon

taurus [48]

Answer:

to find the area of a polygon

formula: area = 1/2 x perimeter x apothem

Step-by-step explanation:

Here is what it means: Perimeter = the sum of the lengths of all the sides. Apothem = a segment that joins the polygon's center to the midpoint of any side that is perpendicular to that side.

3 0

3 years ago

Luna mixes 3/4 cup of orange juice with 3/8 cup of cranberry juice. She gives 5/8 cup of juice to mags. How much is left in luna

Debora [2.8K]

Answer:

The answer is 1/2

Step-by-step explanation:

First you must the least common denominator so that you can add the two fractions. The least common denominator between 3/4 and 3/8, would be 8.

To get 3/4 to the least common denominator, you have to multiply top and bottom by 2. The product of this will be 6/8.

Next you add the two fractions: 6/8 + 3/8 = 9/8

Simplify your improper fraction to get on and 1 eighths or in number form 1 1/8

After this you must subtract how much she gave away so you can use your improper fraction to make it easier and subtract five eighths from nine eighths: 9/8 - 5/8 = 4/8

So, Luna has four eighths of a glass left but first you must simplify your answer. both 4 and 8 divide by 4 to get 1/2.

So Luna's glass only has 1/2 in the cup.

Make sure you be optimistic and see it as half full, instead of the pessimistic view of half empty. Remember to stay positive and share good vibes. Hope this answer helps you.

4 0

3 years ago

For a certain map, the scale shows that 1 cm on the map equals 2.50 km on earth's surface. two towns on the map are 4.75 cm apar

Julli [10]

In this problem, to find the answer we have to setup a series of ratios that relate the scale to real life distance. We know that 1cm = 2.50km, so that ratio would be 1cm/2.5km. For two towns that are 4.75cm apart on the map, we set a ration of 4.75cm/x km, where x is the actual distance. Now we set the ratios equal to each other and solve for x. 1/2.5=4.75/x where x = 4.75*2.5/1 = 11.875 and rounding up we get 11.88 km. The two towns are actually 11.88 km apart from each other.

3 0

3 years ago

8(C - 9) = 6(20 - 12) - 40

krek1111 [17]

Answer:

All Real Numbers

Step-by-step explanation:

Any value of c makes the equation true.

Interval Notation

5 0

3 years ago

Read 2 more answers

I have two quarters anda dime. How much moremoney is needed to make$1.00?​

I have two quarters anda dime. How much moremoney is needed to make$1.00?