What condition must be satisfied by an infinite geometric series in order for its sum to exist?
For each of the following infinite geometric sequences:
Find the common ratio, r.
Find the limiting sum of the series.
2, \, \dfrac{1}{2}, \, \dfrac{1}{8}, \, \dfrac{1}{32}, \ldots
125, 25, 5, 1, \ldots
16, - 8, 4, - 2, \ldots
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
Consider the infinite geometric series: 5 + \sqrt{5} + 1 + \ldots
Find the common ratio, r, expressing your answer with a rational denominator.
Find the limiting sum, expressing your answer with a rational denominator.
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.
For a particular geometric sequence, a = 9 and S_{\infty} = \dfrac{45}{4}.
Find r, the common ratio.
Find the first 3 terms in the sequence.
Consider a geometric series in which all terms are positive that has first term a and common ratio r.
The sum of the infinite series is 243. Form an equation for a in terms of r.
The sum of the first 4 terms is 240. Form another equation for a in terms of r.
Hence, find the value of r.
Hence, find the value of a.
Find the first three terms of the sequence.
The limiting sum of the infinite sequence 1, \, 4x, \, 16 x^{2}, \ldots is 5. Find the value of x.
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.
Consider the recurring decimal 0.57575 \ldots in the form 0.57 + 0.0057 + \ldots Rewrite the recurring decimal as a fraction.
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 a and the common ratio r 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 a and the common ratio r of the sequence formed from these first three terms.
Calculate the infinite sum of the sequence 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
0.12121212 \ldots
0.125125125 \ldots
Use geometric series to rewrite the following recurring decimals as fractions:
1.\dot{3}\dot{2}
0.3\dot{8}
0.\dot{6}\dot{0}
0.1\dot{2}
1.\dot{4}\dot{1}\dot{2}
0.0\dot{1}\dot{5}
2.8\dot{0}\dot{5}
0.10\dot{3}
The perimeter of a triangle is x \text{ cm}. A second triangle is formed by joining the midpoints of the three sides of the original triangle, so that the lengths of the sides are half the lengths of the sides of the original triangle. A third, fourth and fifth triangle are formed in the same way (ie by joining the midpoints of the three sides of the preceding triangle). The perimeter of the 6th triangle is 7 \text{ cm}.
For each new triangle that is formed, describe what is happening to the perimeter.
Find the value of x.
Find the sum of the perimeters of the first 5 triangles.
Find the sum of an infinite number of triangles created in this way.
The annual output of a coal mine decreases by 12 \% each year. The output in the first year is 211\,000 \text{ m}^{3}.
Find the total amount of output in the first 9 years, to two decimal places.
Assuming there is always enough coal in the mine for the plant's operations, what is the limit to this plant's total output production? Round your answer to the nearest \text{m}^{3}.
A new factory in a small town has an annual payroll of \$7 million. It is expected that 65\% of this money will be spent in the town by factory personnel. The people in the town who receive this money are expected to spend 65\% of what they receive in the town, and so on.
Calculate the total of all this spending on the town each year. Round your answer to the nearest dollar.