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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

If the angles of a triangle are 10 45 and 23 what is the perimerter of the triangle

djverab [1.8K]

Answer:

No Solutions

Step-by-step explanation:

In a triangle, the sum of the angles has to be 180 degrees. It also is impossible to find the length of sides without at least one side, since the range of lengths is practically infinite. There are no solutions to this problem.

3 0

3 years ago

For f(x) = 3x - 12 and g(x) = 2x^2 + 7, find f(g(-5)).

Soloha48 [4]

Answer:

159

Step-by-step explanation:

To evaluate f(g(- 5)), first evaluate g(- 5) then substitute this value into f(x)

g(- 5) = 2(- 5)² + 7 = 50 + 7 = 57

f(57) = (3 × 57) - 12 = 171 - 12 = 159

6 0

3 years ago

What is it the formula for calculating volume of a rectangular prism?

taurus [48]

Answer:

idk

Step-by-step explanation: took test

6 0

4 years ago

Read 2 more answers

Pedro cut a sheet of poster board into 10 equal parts. His brother used some of the poster board and now 8/10 is left. Pedro wan

yawa3891 [41]

Answer: 8

Step-by-step explanation:

Given: Pedro cut a sheet of poster board into 10 equal parts.

let x be the total sheet.

then, one part of sheet= $\frac{1}{10}$ of the sheet

Now, His brother used some of the poster board and now 8/10 is left.

Therefore, if Pedro wants to make a sign from each remaining part of the poster board.

then, The number of signs he can make= $\frac{8}{10}\div \frac{1}{10}=\frac{8}{10}\times\frac{10}{1}=8$

Hence, Pedro can make 8 signs.

4 0

4 years ago

Read 2 more answers