Find the percent increase from 138 to 355. Round your answer to the nearest tenth.

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

If x + t =a and x = 3 then why am i so short

agasfer [191]

Good question, I ask myself the same thibg everyday

7 0

4 years ago

HELPPPP PLEASE HELP ME WITH BASIC MATH I REALLY APPRECIATE IT ((WILL MARK AS BRAINLIEST))

Yakvenalex [24]

Answer:

3) 522

4) 468

Step-by-step explanation:

hello!

3) in the question, it says the cross section is a square, and the area of PQRS is 81.

for the square, we know that the length and height have to be the same, so if the area is 81, then we know the sides are both 9 because 9*9 is 81 and they're the same numbers multiplied. (same height and length)

with the information that all 4 sides of the square is 9 and the other side is labeled 10, we can figure out the rest.

both squares : 81+81=162

both 10 by 9 sides: 90+90=180

both 10 by 9 sides (top and bottom): 90+90=180

total surface area=162+180+180=522

4) for this problem, we can see that the triangle is a right triangle.

to solve the area of the triangle, we can do (9*12)/2=108/2=54.

there are two sides with triangles, so we double 54 (54*2=108)

9 by 10 side: 90

10 by 12 side: 120

now for the last slide, theres no information on the length. the formula for the diagonal is

a^2+b^2=c^2

which is

9^2+12^2=81+144=255=15^2

with this, the area of the rectangular side gives us 10*15 which is 150.

adding it together, we get

108+90+120+150=468.

hope this helps!

7 0

3 years ago

12). In 1991, there were an estimated 235 fish in a lake. In 1996, there were 134

weqwewe [10]

Answer:bless you

Step-by-step explanation:

4 0

3 years ago

a local little League has a total of 60 players of whom 20%are left handed .how many left handed players are there?

valentinak56 [21]

Answer:

Step-by-step explanation:

left handed players : 20% of 60

turn ur percent to a decimal...." of " means multiply

0.20 * 60 = 12 are left handed

7 0

4 years ago

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Find the percent increase from 138 to 355. Round your answer to the nearest tenth.​

Find the percent increase from 138 to 355. Round your answer to the nearest tenth.