Please help urgent ❤️❤️❤️Pl

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

Cho tam giác abc có bm,cn là đường trung tuyến . Nếu bm = cn chứng minh tam giác abc cân . Từ đó rút ra định lý trong một tam gi

chubhunter [2.5K]

6251=6635277262562748

4 0

3 years ago

The sum of two numbers is 30. One number is 4 more than twice the other. Find the numbers.

DENIUS [597]

Answer:

11, 25

Step-by-step explanation:

n is a number

2n + 3 is the other number

n + 2n + 3 = 36

4 0

3 years ago

Could someone help me with number 5 because I suck at perimeters

N76 [4]

You might need to use the distance formula

for the distance betwen (x1,y1) and (x2,y2) is
$D= \sqrt{(x1-x2)^2+(y1-y2)^2}$

so find disance between (-3,1) and (-1,3)
and (-1,3) and (3,1)
so
if we did input them
we would get $6+2 \sqrt{2}+2 \sqrt{5}$ which is the peirmiter

6 0

4 years ago

You run a hotel with 1465 rooms. You currently have 296 rooms reserved by individuals and 651 rooms reserved by groups for a par

Tanya [424]

Answer:

Hotel is over booked by 140 rooms

Step-by-step explanation:

Total number of hotel rooms = 1465

number reserved by individuals = 296

number reserved by groups = 651

booking rates = 67% for individual , 56% for groups

<u>Determine if you are under booked or overbooked for the weekend </u>

let X represent the number of rooms booked by individuals

and Y represent the number of rooms booked by groups

number of reserved room for individuals

= 67% of X = 296

X = 67% + 33%

= 296 + ( (296 / 67 ) * 33 )

= 296 + 146 ≈ 442

number of rooms reserved for groups

= 56% of Y = 651

Y =( 651 / 56 ) * 100

≈ 1163

Total rooms booked = X + Y

= 442 + 1163 = 1605

<u>To determine if under booked or over booked </u>

Total rooms booked - Total number of hotel rooms

= 1605 - 1465

= 140 ( Your Hotel is over booked )

4 0

3 years ago