What is the sum from n=3 to infinity of 3/4*1/(4^n) ?

The sum from n=3 to infinity of 3/4*1/(4^n) is 1/64

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sum from n=3 to infinity of 3/4*1/(4^n)

n = 3 \sum \infty \frac{3}{4} \cdot \frac{1}{4 ^{n}}

\frac{1}{64}

Decimal

0.015625

Solution steps

n = 3 \sum \infty \frac{3}{4} \cdot \frac{1}{4 ^{n}}

Apply the constant multiplication rule:

\sum c \cdot a_{n} = c \cdot \sum a_{n}

= \frac{3}{4} \cdot n = 3 \sum \infty \frac{1}{4 ^{n}}

n = k \sum \infty (a_{n}) = n = 0 \sum \infty (a_{n}) - a_{0} - a_{1} - \dots a_{k - 1}

= \frac{3}{4} (n = 0 \sum \infty \frac{1}{4 ^{n}} - \frac{1}{4 ^{0}} - \frac{1}{4 ^{1}} - \frac{1}{4 ^{2}})

n = 0 \sum \infty \frac{1}{4 ^{n}} = \frac{4}{3}

= \frac{3}{4} (\frac{4}{3} - \frac{1}{4 ^{0}} - \frac{1}{4 ^{1}} - \frac{1}{4 ^{2}})

Simplify

\frac{3}{4} (\frac{4}{3} - \frac{1}{4 ^{0}} - \frac{1}{4 ^{1}} - \frac{1}{4 ^{2}}) : \frac{1}{64}

= \frac{1}{64}

What is the sum from n=3 to infinity of 3/4*1/(4^n) ?
The sum from n=3 to infinity of 3/4*1/(4^n) is 1/64