Question

If the population of a town is p in the beginning of any year then it becomes 3+2p in the beginning of the next year. If the population in the beginning of 2019 is 1000, then the population in the beginning of 2034 will be

• A. (997)214 + 3
• B. (1003)15 + 6
• C. (1003)215 - 3
• D. (997)15 - 3

Answer 1

The Correct Option is C

From the given data: 

Year	2019	2020	2021	2022
Population	\( p \)	\( 2p + 3 \)	\( 2(2p + 3) + 3 \)	\( 2[2(2p + 3) + 3] + 3 \)

 

Simplifying the population expressions:

Year	2019	2020	2021	2022
Population	\( p \)	\( 2p + 3 \)	\( 4p + 9 \)	\( 8p + 21 \)

 

The pattern shows that the population is increasing as:

\[ p, 2^1p + 3, 2^2p + 3(1 + 2), 2^3p + 3(1 + 2 + 4), \dots \]

We can see that each year the population is being updated as follows:

• The coefficient of \( p \) doubles each year: \( 2^0p, 2^1p, 2^2p, 2^3p, \dots \)
• The additive term follows a pattern: \( 3 \times (2^0 + 2^1 + 2^2 + \dots) \)

Now, let’s analyze the year 2034.

2034 is 15 years after 2019, so it is the 16^th term in the sequence.

First part: The multiplicative part of \( p \) is:

\[ 2^{15}p \]

Second part: The additive constant is based on the sum:

\[ 3(2^0 + 2^1 + 2^2 + \dots + 2^{14}) \]

This is a geometric series with 15 terms. The sum is:

\[ \sum_{k=0}^{14} 2^k = 2^{15} - 1 \]

So, the additive part is:

\[ 3(2^{15} - 1) \]

Final expression for population in 2034:

\[ \text{Population} = 2^{15}p + 3(2^{15} - 1) \]