114k views
0 votes
Find the sum of the series 9 +3+1+...+1/27​

User Minho
by
7.8k points

1 Answer

5 votes

Answer:


(364)/(27)

Explanation:

A geometric sequence has a constant ratio (multiplier) between each term, so each term is multiplied by the same number, whereas an arithmetic sequence has a constant difference between each term, so the difference between each term is the same.

Given series:

  • 9 + 3 + 1 + ... + 1/27​

As each term of the given sequence is a third of the previous term, the given series is a geometric series. This means the common ratio (r) is 1/3.

From inspection of the given series, the first term (a) is 9.

To find the value of n for the nth term 1/27, simply divide each term by 3:


a_1=9


a_2=(a_1)/(3)=(9)/(3)=3


a_3=(a_2)/(3)=(3)/(3)=1


a_4=(a_3)/(3)=(1)/(3)


a_5=(a_4)/(3)=((1)/(3))/(3)=(1)/(9)


a_6=(a_5)/(3)=((1)/(9))/(3)=(1)/(27)

Therefore, as the 6th term in the sequence is 1/27, we need to find the sum of the series of the first 6 terms. To do this, use the geometric series formula.


\boxed{\begin{minipage}{7 cm}\underline{Sum of the first $n$ terms of a geometric series}\\\\$S_n=(a(1-r^n))/(1-r)$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\ \phantom{ww}$\bullet$ $r$ is the common ratio.\\\end{minipage}}

The given values are:

  • a = 9
  • r = 1/3
  • n = 6

Substitute these values into the formula to find the sum of the first 6 terms:


\implies S_n=(a(1-r^n))/(1-r)


\implies S_6=(9\left(1-(1)/(3)^6\right))/(1-(1)/(3))


\implies S_6=(9\left(1-(1)/(729)\right))/((2)/(3))


\implies S_6=(9\left((728)/(729)\right))/((2)/(3))


\implies S_6=((728)/(81))/((2)/(3))


\implies S_6=(728)/(81) \cdot (3)/(2)


\implies S_6=(2184)/(162)


\implies S_6=(364)/(27)


\hrulefill

Alternative method

As there are only 6 terms to sum, an alternative method would be to simply add the 6 terms:


\implies S_6=9+3+1+(1)/(3)+(1)/(9)+(1)/(27)

Rewrite all the numbers so that they are fractions with the same denominator of 27:


\implies S_6=(243)/(27)+(81)/(27)+(27)/(27)+(9)/(27)+(3)/(27)+(1)/(27)

As the denominators are the same, simply add the numerators:


\implies S_6=(243+81+27+9+3+1)/(27)=(364)/(27)

User Mark Avenius
by
9.0k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories