For each of the following infinite geometric sequences:
Find the common ratio, r.
Find the limiting sum of the series.
16, - 8, 4, - 2, \ldots
For a particular geometric sequence, t_1 = 2 and S_{\infty} = 4.
Find the common ratio, r of the series.
Write the first three terms in the geometric progression.
Consider the recurring decimal 0.2222 \ldots in the form \dfrac{2}{10} + \dfrac{2}{100} + \dfrac{2}{1000} + \dfrac{2}{10\,000} + \ldots Rewrite the recurring decimal as a fraction.
For each of the following recurring decimals:
Rewrite the decimal as a geometric series.
Express the decimal as a fraction.
0.6666 \ldots
0.06666 \ldots
Consider the infinite sequence 1, x - 7, \left(x - 7\right)^{2}, \left(x - 7\right)^{3}, \ldots
For what range of values of x does the infinite series have a limiting sum?
Consider the recurring decimal 0.57575 \ldots in the form 0.57 + 0.0057 + \ldots Rewrite the recurring decimal as a fraction.
Consider the sum 0.25 + 0.0025 + 0.000025 + \text{. . .}
Express the sum as a decimal.
Hence, express the sum as a fraction.
The limiting sum of the infinite sequence 1, 4 x, 16 x^{2}, \ldots is 5. Solve for the value of x.
Consider the infinite geometric series: 6 + 2 + \dfrac{2}{3} + \dfrac{2}{9} + \ldots
Find the number of terms, n, that would be required to give a sum of \dfrac{177\,146}{19\,683}.
Find the sum of an infinite number of terms.
Consider the infinite geometric series: 5 + \sqrt{5} + 1 + \ldots
Find the exact value of the common ratio, r.
Find the exact value of the limiting sum.
Find the limiting sum of the infinite series \dfrac{1}{5} + \dfrac{3}{5^{2}} + \dfrac{1}{5^{3}} + \dfrac{3}{5^{4}} + \dfrac{1}{5^{5}} + \ldots
The recurring decimal 0.8888 \ldots can be expressed as a fraction when viewed as an infinite geometric series.
Express the first decimal place, 0.8 as an unsimplified fraction.
Express the second decimal place, 0.08 as an unsimplified fraction.
Hence, using fractions, write the first five terms of the geometric sequence representing 0.8888 \ldots
State the values of the first term t_1 and the common ratio t_1 of this sequence.
Calculate the infinite sum of the sequence as a fraction.
The recurring decimal 0.444444 \ldots can be expressed as a fraction when viewed as an infinite geometric series.
Express the first two decimal places, 0.44, as an unsimplified fraction.
Express the second two decimal places, 0.0044, as an unsimplified fraction.
Express the third two decimal places, 0.000044, as an unsimplified fraction.
Hence, state the values of the first term t_1 and the common ratio r of the sequence formed from these first three terms.
Calculate the infinite sum of the sequence as a fraction.
Evaluate:
\sum_{i=1}^{\infty} 3 \left(\dfrac{1}{4}\right)^{i - 1}
\sum_{i=1}^{\infty} 5 \left( - \dfrac{1}{4} \right)^{i - 1}
\sum_{i=1}^{\infty} \left(0.9\right)^{i}
\sum_{k=1}^{\infty} 9^{ - k }
\sum_{i=1}^{\infty} \dfrac{1}{8} \left( - \dfrac{1}{2} \right)^{i - 1}
\sum_{i=1}^{\infty} - \dfrac{1}{2} \left(\dfrac{5}{7}\right)^{i - 1}
A ball dropped from a height of 21\text{ m} will bounce back off the ground to 50\% of the height of the previous bounce (or the height from which it is dropped when considering the first bounce).
Write a function, y, to represent the height of the nth bounce.
Calculate the height of the fifth bounce. Round your answer correct to two decimal places.
The annual output of a coal mine decreases by twenty percent each year. The output in the first year is 274\,000\text{ m}^3.
Find the total amount of output in the first 8 years, to two decimal places if needed.
Assuming there is always enough coal in the mine for the plant's operations, find the limit to this plant's total output production, to the nearest cubic metre.
When a ball is dropped onto a horizontal surface from a height of 6\text{ m}, it reaches a vertical height of 50\% of the starting height on its first bounce. It continues to reach a height of 50\% of the previous height in each subsequent bounce.
Find the height of the first bounce.
Find the height of the 5th upward bounce, correct to four decimal places.
Calculate the total vertical distance travelled by the ball when it touches the ground after its third upward bounce.
If the ball continues to bounce until it finally stops, how far has it travelled vertically?
When a marble is rolled horizontally on a flat surface it rolls 30\text{ cm} in the first second. It then rolls 60\% of the distance travelled in the previous second, for each subsequent second.
Determine the distance rolled in the 2nd second.
Determine the total distance rolled after 3 seconds, correct to one decimal place.
If the marble continues to roll until it finally stops, how far has it travelled horizontally?