A grade 2 class was playing guess my rule

The sum of the first n terms of a geometric series is 364? The sum of their reciprocals 364/243. If the first term is 1, find n

Afina-wow [57]

If the geometric series has first term $a$ and common ratio $r$ , then its $N$ -th partial sum is

$\displaystyle S_N = \sum_{n=1}^N ar^{n-1} = a + ar + ar^2 + \cdots + ar^{N-1}$

Multiply both sides by $r$ , then subtract $rS_N$ from $S_N$ to eliminate all the middle terms and solve for $S_N$ :

$rS_N = ar + ar^2 + ar^3 + \cdots + ar^N$

$\implies (1 - r) S_N = a - ar^N$

$\implies S_N = \dfrac{a(1-r^N)}{1-r}$

The $N$ -th partial sum for the series of reciprocal terms (denoted by $S'_N$ ) can be computed similarly:

$\displaystyle S'_N = \sum_{n=1}^N \frac1{ar^{N-1}} = \frac1a + \frac1{ar} + \frac1{ar^2} + \cdots + \frac1{ar^{N-1}}$

$\dfrac{S'_N}r = \dfrac1{ar} + \dfrac1{ar^2} + \dfrac1{ar^3} + \cdots + \dfrac1{ar^N}$

$\implies \left(1 - \dfrac1r\right) S'_N = \dfrac1a - \dfrac1{ar^N}$

$\implies S'_N = \dfrac{1 - \frac1{r^N}}{a\left(1 - \frac1r\right)} = \dfrac{r^N - 1}{a(r^N - r^{N-1})} = \dfrac{1 - r^N}{a r^{N-1} (1 - r)}$

We're given that $a=1$ , and the sum of the first $n$ terms of the series is

$S_n = \dfrac{1-r^n}{1-r} = 364$

and the sum of their reciprocals is

$S'_n = \dfrac{1 - r^n}{r^{n-1}(1 - r)} = \dfrac{364}{243}$

By substitution,

$\dfrac{1 - r^n}{r^{n-1}(1-r)} = \dfrac{364}{r^{n-1}} = \dfrac{364}{243} \implies r^{n-1} = 243$

Manipulating the $S_n$ equation gives

$\dfrac{1 - r^n}{1-r} = 364 \implies r (364 - r^{n-1}) = 363$

so that substituting again yields

$r (364 - 243) = 363 \implies 121r = 363 \implies \boxed{r=3}$

and it follows that

$r^{n-1} = 243 \implies 3^{n-1} = 3^5 \implies n-1 = 5 \implies \boxed{n=6}$

5 0

2 years ago

Here is Elena’s work for finding the surface area of a rectangular prism that is 1 foot by 1 foot by 2 feet. She concluded that

Tema [17]

ANSWER PLEASE

A cube of sugar has a mass of 12 grams and a volume of 8 cubic centimeters (cm^3). What is its density? [You may use a calculator.] Choose the BEST answer.

0.67 g/cm^3

0.67

1.5 g/cm^3

1.5

6 0

3 years ago

How many times does 3 go into 92

ollegr [7]

To find if 3 goes int 92

we need to divide 92 by 3

Figure of the division attached

The first part is 3 goes 3 times into 9 leaving remainder 0

We are left with 2 to divide

but since 3<2, we add 0 to the quotient and leave the 2 as the remainder

Hence 3 goes 30 times in 92 leaving a remainder of 2

3 0

3 years ago

Read 2 more answers

The Skyview Atlanta in Atlanta, Georgia is a super-sized Ferris wheel that overlooks the city

katrin [286]

Answer: a. 279.11meters

b. 2513.27 feet

Step-by-step explanation:

Circumference of a circle= 2πr

where r = radius of the circle.

In the picture, there are t

18 spokes, therefore the angle between any two spokes will be:

360° ÷ 18 = 20°

The arc length traversed by passenger will be= radius of the circle × the angle that is traversed by the passenger

radius = 200 feet.

He has traveled from M to J,that subtends 4 × 20 = 80°

= 80π/180 radian = 4π/9radian

The passenger has therefore traveled:

200 × 4π/9 = 800π/9

= (800×3.14)/9

= 279.11meters

Part B.

Full ride is equivalent to 2 × the circumference of the circle= 2 × 2πr

= 4πr

= 4 × 3.14 × 200

= 2512 feet

So,he would travel 4* 7 * 200 feet = 2513.27 feet

4 0

4 years ago

A. Use composition to prove whether or not the functions are inverses of each other. B. Express the domain of the compositions u

Furkat [3]

Answer:

do what I would do just answer what ever and if you get it wrong do it until you get it right thats what my momma tell me to do

Step-by-step explanation:

6 0

3 years ago