Question

In 2007 , US natural gas consumption was 652.9 billion cubic meters. Asian consumption was 447.8 billion cubic meters. \(^{7}\) During the previous decade, US consumption increased by only \(0.14 \%\) a year, while Asian consumption grew by \(5.93 \%\) a year. Assume these rates continue into the future. (a) Give the first four terms of the sequence, \(a_{n}\), giving US consumption of natural gas \(n\) years after \(2006 .\) (b) Give the first four terms of a similar sequence \(b_{n}\) showing Asian gas consumption. (c) According to this model, when will Asian yearly gas consumption exceed US consumption?

Short answer

(a) The first four terms of the US gas consumption sequence \(a_n\) are: \(a_1 = 652.9\) billion cubic meters, \(a_2 \approx 653.8\) billion cubic meters, \(a_3 \approx 654.7\) billion cubic meters, \(a_4 \approx 655.5\) billion cubic meters. (b) The first four terms of the Asian gas consumption sequence \(b_n\) are: \(b_1 = 447.8\) billion cubic meters, \(b_2 \approx 474.2\) billion cubic meters, \(b_3 \approx 502.0\) billion cubic meters, \(b_4 \approx 531.4\) billion cubic meters. (c) According to the model, Asian gas consumption will exceed US consumption in the year 2015, which is 9 years after 2006.

Step-by-step solution

(a) USA Gas Consumption Sequence

To find the first four terms of the US gas consumption sequence \(a_n\), we'll use the initial consumption value in 2007 (652.9 billion cubic meters) and the given percentage increase of 0.14% per year. First, convert the percentage increase to a multiplier: Multiplier = 1 + (percentage increase / 100) = 1 + (0.14 / 100) = 1.0014 Now, create the sequence by multiplying the previous value with the multiplier: \(a_1\) = Initial value in 2007 = 652.9 billion cubic meters \(a_2 = a_1 \cdot 1.0014\) \(a_3 = a_2 \cdot 1.0014\) \(a_4 = a_3 \cdot 1.0014\)

(b) Asian Gas Consumption Sequence

Similarly, to find the first four terms of the Asian gas consumption sequence \(b_n\), we use the initial consumption value in 2007 (447.8 billion cubic meters) and the given percentage increase of 5.93% per year. First, convert the percentage increase to a multiplier: Multiplier = 1 + (percentage increase / 100) = 1 + (5.93 / 100) = 1.0593 Now, create the sequence by multiplying the previous value with the multiplier: \(b_1\) = Initial value in 2007 = 447.8 billion cubic meters \(b_2 = b_1 \cdot 1.0593\) \(b_3 = b_2 \cdot 1.0593\) \(b_4 = b_3 \cdot 1.0593\)

(c) Asian Consumption Exceeding USA Consumption

To find when Asian consumption exceeds US consumption according to this model, we need to find the smallest integer value of \(n\) such that \(b_n > a_n\). To do this, we can either find a general formula for both sequences and solve for \(n\) or compare the values until we find the first instance of \(b_n > a_n\). By comparing the values, we can observe that Asian consumption will exceed US consumption when \(n=9\) years after 2006, which corresponds to the year 2015.

Key concepts

Exponential Growth

Exponential growth refers to the process where the quantity increases at a consistent percentage rate over time. This is different from linear growth where the increase is by a fixed amount each period. In the context of the textbook exercise, both the US and Asian natural gas consumption are growing exponentially, but at different rates.
- For the US, the rate is 0.14% per year, which is quite slow. - For Asia, the growth rate is much higher, at 5.93% per year.
When a quantity grows exponentially, each term in the sequence is derived by multiplying the previous term by a fixed multiplier. This multiplier is calculated from the growth rate. For example, a 0.14% growth translates to a multiplier of 1.0014. This consistent multiplication leads to an accelerating increase in the quantity.

Percentage Increase

Percentage increase is a way to describe how much a number grows, depicted as a percentage of the original amount. This is particularly useful in financial contexts or when comparing growth rates between different sequences.
In the given sequence task, understanding percentage increase is crucial to forming the sequences of natural gas consumption.

• To convert a percentage increase to a multiplier, use this formula: Multiplier = 1 + (percentage increase/100).
• For the US, a 0.14% increase becomes a multiplier of 1.0014.
• For Asia, a 5.93% increase becomes a multiplier of 1.0593.

Applying the multiplier to the initial value gives the next term in the sequence. The cumulative effect over multiple years highlights the benefits of exponential rather than linear growth.

Comparison of Sequences

When comparing sequences of growth, it is important to analyze both their initial values and their growth rates. A sequence with a larger starting value can initially appear stronger, but over time, a sequence with a higher growth rate can surpass it.
In the exercise, the US starts with a higher consumption of 652.9 billion cubic meters compared to Asia's 447.8 billion cubic meters. However, due to the much higher growth rate of 5.93% for Asia, its consumption eventually surpasses that of the US at 0.14% growth rate.

• Initially, the US sequence seems dominant due to its higher start.
• Over time, Asia's sequence grows significantly faster because of the larger percentage increase.

Comparing sequences shows how exponential growth rates can result in a dramatic change of lead even from a lower starting point.

Growth Rates

Growth rates indicate how quickly a quantity increases over time and are crucial in determining future trends. In the exercise, the focus is on comparing how different growth rates affect consumption trends over time.
A higher growth rate suggests a faster increase:

• The US's 0.14% rate results in a slow growth, almost stable.
• Asia's 5.93% rate shows dynamic, rapid growth.

In practice, lower growth rates might suit situations requiring stability, whereas higher rates align with scenarios seeking rapid scaling. The textbook example illustrates how over the span of years, even a small consistent increase — when repeated over time — can result in substantial differences, as seen with Asia overtaking the US gas consumption by the year 2015.