8.05 Geometric series (Enrichment)
Find the sum of the first 5 terms of the geometric sequence defined by the following. Round your answers to two decimal places.
u_1 = 2.187 and r = 1.134
u_1 = - 4.186 and r = - 2.848
Consider the series 5 + \dfrac{5}{2} + \dfrac{5}{4} + \ldots
Find the common ratio, r.
Find the sum of the first 9 terms, correct to one decimal place.
Find the sum of the first 12 terms in the series 4 + 12 + 36 + \ldots
Find the sum of the first 7 terms of the geometric series: 64 + 16 + 4 \ldots
Find the sum of the first 11 terms of the series: 1 - 2 + 4 \ldots
Find the sum of the first 5 terms of the series: 24 - 12 + 6 \ldots
Leave your answer correct to the nearest whole number.
Find the sum of the first 7 terms of the series: 8 + 1 + \dfrac{1}{8} \ldots
Round your answer to three decimal places.
The first 3 terms of a sequence are x, \, \dfrac{3}{4} x^{2}, \, \dfrac{9}{16} x^{3}.
Write a simplified expression for the sum of the first n terms of this sequence.
Evaluate the sum when x = 4 and n = 8.
Consider the series \dfrac{1}{4} - \dfrac{1}{5} + \dfrac{1}{16} - \dfrac{1}{25} + \dfrac{1}{64} - \dfrac{1}{125} + \ldots
Form an expression for the sum of the first 2 n terms of the series.
The sum of the first 7 terms of a geometric series is 2186, and the common ratio is \dfrac{1}{3}.
Find u_1, the first term in the series.
The sum of the first 6 terms of a geometric series is 9 times the sum of its first 3 terms.
Find r, the common ratio.
Find n, the number of terms of the following geometric series:
The sum of n terms in the geometric series 1 + 4 + 16 + \ldots is 89\,478\,485.
The sum of n terms in the geometric series 16 + 4 + 1 + \ldots is \dfrac{5461}{256}.
Consider the series 64 + 16 + 4 + \ldots + \dfrac{1}{1024}.
Find n, the number of terms in the series.
Find the sum of the series, correct to three decimal places.
Consider the series 4 - 12 + 36 - \ldots - 708\,588.
Find n, the number of terms in the series.
Find the sum of the series.
Consider the geometric series 20, - 4, \dfrac{4}{5}, \ldots
Find the number of terms n, that have a sum of \dfrac{10\,416}{625}.
Consider the series 5 + 10 + 20 + \ldots.
Find the sum of the first n terms of the series in terms of n.
Find n, the total number of terms if the sum is equal to 2555.
Find how many terms are less than 50\,000.
Find the first sum of n terms that is greater than 50\,000.
Prove that the sum of the first n terms is always 5 less than the \left( n + 1 \right)th term.
The powers of 3 greater than 1 form a geometric progression 3, 9, 27, \ldots
Find how many terms must be added for the sum to exceed 10^{25}.
Suppose you save \$1 the first day of a month, \$2 the second day, \$4 the third day, \$8 the fourth day, and so on. That is, each day you save twice as much as you did the day before.
How much will your total savings be for the first 13 days?
How much will your total savings be for the first 29 days?
Average annual salaries are expected to increase by 4 \% each year. If the average annual salary this year is found to be \$40\,000.
Calculate the expected average annual salary in 6 years.
This year, Vincent starts at a new job in which he will receive the average annual salary for each year of his employment. Over the coming 6 years (including this year) he plans to save half of each year’s annual salary.
Calculate his total savings over these 6 years.
This year, 600 people are expected to enter the workforce as registered nurses. This number is expected to increase by 4\% next year, and increase by the same percentage every year after that.
Calculate the following, rounding your answers to the nearest whole number:
The number of nurses expected to enter the workforce between six and seven years from now.
The number of nurses expected to enter the workforce over the next six years.
A conveyor belt is being used to remove materials from a quarry. Every 45 minutes, the conveyor belt empties out \dfrac{1}{5} of whatever material remains in the quarry. The quarry initially holds 19\,500 \text{ m}^{3} of materials.
How much material has been pumped out after 135 minutes?
How much material is left in the dam after 135 minutes?