Geometric sequence of infinite

Geometry Level 1

1 6 , 1 6 2 , 1 6 3 , 1 6 4 , 1 6 5 , ⋯ \large \frac{1}{6},\frac{1}{6^{2}}, \frac{1}{6^{3}}, \frac 1{6^4}, \frac 1{6^5}, \cdots

What the sum of the infinite sequence above?

1 36 \frac 1{36} 1 25 \frac 1{25} 1 5 \frac 15 1 6 \frac 16

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Lorenz Navales by lorenz navales , 21, Philippines

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3 solutions

Chew-Seong Cheong
Aug 30, 2016

Relevant wiki: Geometric Progression Sum

S = 1 6 + 1 6 2 + 1 6 3 + 1 6 4 + 1 6 5 + . . . = ∑ n = 1 ∞ ( 1 6 ) n This is a sum of infinite GP. = 1 6 ⋅ 1 1 − 1 6 = 1 6 ⋅ 6 5 = 1 5 \begin{aligned} S & = \frac 16 + \frac 1{6^2} + \frac 1{6^3} + \frac 1{6^4} + \frac 1{6^5} + ... \\ & = \sum_{n=1}^\infty \left(\frac 16\right)^n & \small \color{#3D99F6}{\text{This is a sum of infinite GP.}} \\ & = \frac 16 \cdot \frac 1{1-\frac 16} = \frac 16 \cdot \frac 65 = \boxed{\dfrac 15} \end{aligned}

Another solution

S = 1 6 + 1 6 2 + 1 6 3 + 1 6 4 + 1 6 5 + . . . = 1 6 + 1 6 ( 1 6 + 1 6 2 + 1 6 3 + 1 6 4 + . . . ) = 1 6 + 1 6 S ⟹ 5 6 S = 1 6 S = 1 5 \begin{aligned} S & = \frac 16 + \frac 1{6^2} + \frac 1{6^3} + \frac 1{6^4} + \frac 1{6^5} + ... \\ & = \frac 16 + \frac 16 \left( \frac 16 + \frac 1{6^2} + \frac 1{6^3} + \frac 1{6^4} + ... \right) \\ & = \frac 16 + \frac 16 S \\ \implies \frac 56 S & = \frac 16 \\ S & = \boxed{\dfrac 15} \end{aligned}

Note: In fact, we can find the general formula S ∞ = 1 1 − r S_\infty = \dfrac 1{1-r} this way.

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how to solve if the question was for 5/6

abhishek alva - 4 years, 9 months ago

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The general formula for sum of infinite GP is:

∑ n = 1 ∞ r n = r 1 − r ⟹ ∑ n = 1 ∞ ( 3 6 ) n = 5 6 1 − 5 6 = 5 \begin{aligned} \sum_{n=\color{#D61F06}{1}}^\infty r^n & = \frac r{1-r} \\ \implies \sum_{n=1}^\infty \left(\frac 36 \right)^n & = \frac {\frac 56}{1-\frac 56} = \boxed{5} \end{aligned}

You should refer to the wiki: Geometric Progression Sum I have provided in the solution to learn more.

Chew-Seong Cheong - 4 years, 9 months ago
Lorenz Navales
Aug 30, 2016

1 6 , 1 6 2 , 1 6 3 , 1 6 4 , 1 6 5 , ⋯ \large \frac{1}{6},\frac{1}{6^{2}}, \frac{1}{6^{3}}, \frac 1{6^4}, \frac 1{6^5}, \cdots

a 1 a_{1} = 1 6 \frac{1}{6}
r = 1 6 \frac{1}{6}
and the formula is S n S_{n} = a 1 a_{1} /1-r

S n S_{n} = 1 6 \frac{1}{6} /1- 1 6 \frac{1}{6}

S n S_{n} = 1 6 \frac{1}{6} / 5 6 \frac{5}{6}

S n S_{n} = 1 5 \frac{1}{5}

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r = a 2 a 1 = 1 36 1 6 = 1 36 × 6 1 = 1 6 r=\dfrac{a_2}{a_1}=\dfrac{\dfrac{1}{36}}{\dfrac{1}{6}}=\dfrac{1}{36}\times \dfrac{6}{1}=\dfrac{1}{6}

s = a 1 1 − r = 1 6 1 − 1 6 = 1 5 s=\dfrac{a_1}{1-r}=\dfrac{\dfrac{1}{6}}{1-\dfrac{1}{6}}=\boxed{\dfrac{1}{5}}

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