sum((1/2)^n,n=1,4) — step-by-step solution

1) Let's calculate:

\sum_{n = 1}^{4} {(\frac{1}{2})}^{n}

For

\sum_{k = a}^{b} c^{k + d}

, you can use the formula:

c^{a + d} \cdot \frac{c^{b - a + 1} - 1}{c - 1}

ⅰ

\sum_{n = 1}^{4} {(\frac{1}{2})}^{n}

\frac{1}{2} \cdot \frac{{(\frac{1}{2})}^{4 - 1 + 1} - 1}{\frac{1}{2} - 1}

\frac{1}{2} \cdot \frac{{(\frac{1}{2})}^{4 - 1 + 1} - 1}{\frac{1}{2} - 1}

ⓘ To multiply fractions, multiply the numerators together and the denominators together:

\frac{1}{2} \cdot \frac{{(\frac{1}{2})}^{4 - 1 + 1} - 1}{\frac{1}{2} - 1}

\frac{1 \cdot ({(\frac{1}{2})}^{4 - 1 + 1} - 1)}{2 \cdot (\frac{1}{2} - 1)}

\frac{{(\frac{1}{2})}^{4 - 1 + 1} - 1}{2 \cdot (\frac{1}{2} - 1)}

■The exponent is computed before the power:

- 1 + 1

0

■To add fractions, put them over the same denominator, then add the numerators:

\frac{1}{2} - 1

\frac{1 - 1 \cdot 2}{2}

\frac{{(\frac{1}{2})}^{4 + 0} - 1}{2 \cdot \frac{1 - 1 \cdot 2}{2}}

ⓘ Adding 0 changes nothing:

4 + 0

4

■Multiplications are done before additions and subtractions:

1 \cdot 2

2

\frac{{(\frac{1}{2})}^{4} - 1}{2 \cdot \frac{1 - 2}{2}}

ⓘ Multiplying a fraction by a number multiplies its numerator:

2 \cdot \frac{1 - 2}{2}

\frac{2 \cdot (1 - 2)}{2}

■Parentheses are computed first:

1 - 2

-1

■Multiplying by -1 gives the opposite:

2 \cdot (-1)

-2

\frac{{(\frac{1}{2})}^{4} - 1}{- \frac{2}{2}}

ⓘ The sign of a fraction goes in front of the bar, whether it comes from the numerator or the denominator:

\frac{{(\frac{1}{2})}^{4} - 1}{- \frac{2}{2}}

- \frac{{(\frac{1}{2})}^{4} - 1}{\frac{2}{2}}

- \frac{{(\frac{1}{2})}^{4} - 1}{\frac{2}{2}}

ⓘ The minus sign of the fraction goes into the numerator or the denominator, changing the sign of each of its terms:

- \frac{{(\frac{1}{2})}^{4} - 1}{\frac{2}{2}}

\frac{- {(\frac{1}{2})}^{4} + 1}{\frac{2}{2}}

\frac{- {(\frac{1}{2})}^{4} + 1}{\frac{2}{2}}

ⓘ Dividing is multiplying by the reciprocal:

\frac{- {(\frac{1}{2})}^{4} + 1}{\frac{2}{2}}

(- {(\frac{1}{2})}^{4} + 1) \cdot \frac{2}{2}

■We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{2}{2}

1

(- {(\frac{1}{2})}^{4} + 1) \cdot 1

- {(\frac{1}{2})}^{4} + 1

ⓘ The power of a quotient is the quotient of the powers, (a/b)^n = a^n/b^n:

- {(\frac{1}{2})}^{4}

- \frac{1}{2^{4}}

- \frac{1}{2^{4}} + 1

ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

2^{4}

2 \cdot 2 \cdot 2 \cdot 2

16

- \frac{1}{16} + 1

ⓘ To add fractions, put them over the same denominator, then add the numerators:

- \frac{1}{16} + 1

\frac{- 1 + 16}{16}

\frac{- 1 + 16}{16}

ⓘ The numerator and the denominator are computed before the division:

- 1 + 16

15

▶Result:

\frac{15}{16}

2) The numeric value of:

\frac{15}{16}

equals:

▷

0.9375