Question

an ancient treasure, discovered 10 years ago, had an initial value of $1,000,000. the value of this treasure has grown every year since its discovery by $25,000.

the explicit formula that can be used to model the value, $a_n$, of this ancient treasure after $n$ years, is \boxed{}. today, 10 years after discovery, the treasure has a value of \boxed{}.

\begin{tabular}{|c|c|}hline $a_n = 1025000 + (n - 1)(25000)$ & $a_n = 25000 + (n - 1)(1000000)$ \hline $a_n = a_{n - 1} + 25000$ & $a_{n - 1} = a_n + 25000$ quad $$1,025,000$ \hline $$1,175,000$ & LXB11,300,000$ \hlineend{tabular}

Explanation:

Step1: Identify the sequence type

This is an arithmetic sequence where the initial value \( a_1 = 1000000 \) and the common difference \( d = 25000 \). The explicit formula for an arithmetic sequence is \( a_n=a_1+(n - 1)d \). Substituting the values, we get \( a_n = 1000000+(n - 1)(25000) \)? Wait, no, wait. Wait, when \( n = 1 \), it's the initial value. Wait, maybe the first term is at \( n = 0 \)? Wait, no, the problem says "after \( n \) years". Wait, the initial value is at discovery (0 years? Or 1 year?). Wait, the treasure was discovered 10 years ago, initial value \( \$1,000,000 \). Then each year it grows by \( \$25,000 \). So the explicit formula for an arithmetic sequence is \( a_n=a_1+(n - 1)d \), but if \( n = 1 \) is the first year after discovery, then \( a_1 = 1000000 \), \( d = 25000 \), so \( a_n=1000000+(n - 1)25000 \). But let's check the options. Wait, one of the options is \( a_n = 25000+(n - 1)(1000000) \) which is wrong, because the common difference is 25000, not 1000000. Another option is \( a_n = 1025000+(n - 1)(25000) \), no. Wait, maybe the initial term is at \( n = 0 \), so \( a_n=1000000 + 25000n \). But let's check the value after 10 years. After 10 years, the value is \( 1000000+25000\times10=1000000 + 250000=1250000 \)? Wait, no, wait the options include \( \$1,250,000 \). Wait, but let's re - examine the formula. Wait, maybe the first term \( a_1 = 1000000 \) (at year 1), then \( a_n=a_1+(n - 1)d \). For \( n = 10 \), \( a_{10}=1000000+(10 - 1)\times25000=1000000 + 225000=1225000 \)? No, that's not in the options. Wait, maybe the initial value is at year 0, so \( a_n=1000000+25000n \). Then at \( n = 10 \), \( a_{10}=1000000+25000\times10 = 1250000 \), which is one of the options. Now, the explicit formula: the arithmetic sequence explicit formula is \( a_n=a_1+(n - 1)d \), but if \( a_1 = 1000000 \), \( d = 25000 \), then \( a_n=1000000+(n - 1)25000=1000000 - 25000+25000n=975000 + 25000n \), which is not in the options. Wait, maybe the formula is \( a_n=25000+(n - 1)1000000 \) which is wrong. Wait, another option: \( a_n=a_{n - 1}+25000 \) is a recursive formula, not explicit. So the explicit formula should be \( a_n = 1000000+25000n \)? But that's not in the options. Wait, wait the options for the formula: let's check again. The options are:

1. \( a_n = 1025000+(n - 1)(25000) \)
2. \( a_n = 25000+(n - 1)(1000000) \)
3. \( a_n=a_{n - 1}+25000 \) (recursive)
4. \( a_{n - 1}=a_n+25000 \) (wrong, since it's decreasing)

Wait, maybe the problem considers the initial value at \( n = 1 \) as \( 1000000 \), and the formula is \( a_n=1000000+(n - 1)25000 \), but when we expand that, \( a_n=1000000 - 25000+25000n=975000 + 25000n \). But that's not matching. Wait, maybe the first term is at \( n = 0 \), so \( a_0 = 1000000 \), then \( a_n=1000000+25000n \). For \( n = 10 \), \( a_{10}=1000000+25000\times10 = 1250000 \), which is an option. Now, the explicit formula: among the options, the correct explicit formula? Wait, no, the options for the formula: maybe I made a mistake. Wait, the problem says "the explicit formula that can be used to model the value, \( a_n \), of this ancient treasure after \( n \) years". Let's think again. The sequence is arithmetic with first term \( a_1 = 1000000 \) (at year 1) and common difference \( d = 25000 \). So the explicit formula is \( a_n=a_1+(n - 1)d=1000000+(n - 1)25000 \). But when we look at the options, none of the formula options seem to match this. Wait, wait, one of the formula options is \( a_n = 25000+(n - 1)(1000000) \) which is wrong. Another is \( a_n = 1025000+(n - 1)(25000) \), which would mean \( a_1 = 1025000 \), which is not the initial value. Wait, maybe the problem has a different interpretation. Maybe the "initial value" is at \( n = 1 \), and the formula is \( a_n=1000000+25000(n - 1) \), which is \( a_n=975000+25000n \). But that's not in the formula options. Wait, maybe the formula is \( a_n = 25000n+1000000 \), but that's not in the formula options. Wait, the formula options: the only explicit formula (non - recursive) is \( a_n = 25000+(n - 1)(1000000) \) (wrong) or \( a_n = 1025000+(n - 1)(25000) \) (wrong) or the recursive \( a_n=a_{n - 1}+25000 \). But the question asks for the explicit formula. Wait, maybe the problem has a typo, or I'm misinterpreting. Alternatively, maybe the initial value is at \( n = 0 \), so \( a_0 = 1000000 \), and the formula is \( a_n=1000000+25000n \), which is not in the options. Wait, but the value after 10 years: initial value \( 1,000,000 \), each year grows by \( 25,000 \), so after 10 years, the growth is \( 10\times25,000 = 250,000 \), so total value is \( 1,000,000+250,000 = 1,250,000 \). So the value after 10 years is \( \$1,250,000 \). Now, for the explicit formula, maybe the correct one is \( a_n = 1000000+25000n \), but since that's not in the formula options, maybe the problem considers the formula as \( a_n = 25000+(n - 1)1000000 \) (wrong) or the recursive formula is not explicit. Wait, maybe the explicit formula is \( a_n = 1000000+25000n \), but since it's not in the options, maybe the problem has a mistake. But based on the value, after 10 years, it's \( 1,000,000+10\times25,000=1,250,000 \). And for the formula, maybe the intended explicit formula is \( a_n = 1000000+25000n \), but since that's not in the options, maybe the formula option is a trick. Wait, the formula options: the only non - recursive formula (explicit) is \( a_n = 25000+(n - 1)1000000 \) (wrong) or \( a_n = 1025000+(n - 1)25000 \) (wrong) or the recursive ones. Wait, maybe the problem considers the explicit formula as \( a_n = 1000000+25000n \), but since it's not in the options, maybe the answer for the formula is none, but that can't be. Wait, no, maybe I made a mistake in the sequence. Let's re - express:

The value at year 0 (discovery) is \( 1,000,000 \).

Value at year 1: \( 1,000,000 + 25,000=1,025,000 \)

Value at year 2: \( 1,025,000+25,000 = 1,050,000 \)

So the explicit formula is \( a_n=1,000,000+25,000n \), where \( n \) is the number of years after discovery.

So for \( n = 10 \), \( a_{10}=1,000,000+25,000\times10 = 1,250,000 \)

Now, looking at the formula options:

- \( a_n = 1025000+(n - 1)(25000) \): If \( n = 1 \), \( a_1=1025000 \), which is the value at year 1, so this formula is \( a_n=a_1+(n - 1)d \), where \( a_1 = 1025000 \), \( d = 25000 \). But the initial value (at \( n = 0 \)) would be \( 1025000-25000 = 1,000,000 \), which matches. So this formula is correct if \( n \) starts at 1 (year 1). So \( a_n=1025000+(n - 1)(25000) \) is the explicit formula (arithmetic sequence formula with \( a_1 = 1025000 \), \( d = 25000 \)).

Then, for \( n = 10 \) (10 years after discovery, so \( n = 10 \) in this formula), \( a_{10}=1025000+(10 - 1)\times25000=1025000 + 225000=1,250,000 \)

Yes, that works. So the explicit formula is \( a_n = 1025000+(n - 1)(25000) \) and the value after 10 years is \( \$1,250,000 \)

Step1: Determine the explicit formula

The value of the treasure forms an arithmetic sequence. The initial value (at \( n = 1 \), 1 year after discovery) is \( a_1=1000000 + 25000=1025000 \) (since at year 0 it's 1,000,000, year 1 is 1,025,000), and the common difference \( d = 25000 \). The explicit formula for an arithmetic sequence is \( a_n=a_1+(n - 1)d \). Substituting \( a_1 = 1025000 \) and \( d = 25000 \), we get \( a_n=1025000+(n - 1)(25000) \)

Step2: Calculate the value after 10 years

We use the explicit formula with \( n = 10 \):

\( a_{10}=1025000+(10 - 1)\times25000 \)

First, calculate \( (10 - 1)\times25000=9\times25000 = 225000 \)

Then, \( a_{10}=1025000+225000=1250000 \)

Answer:

The explicit formula is \( \boldsymbol{a_n = 1025000+(n - 1)(25000)} \) and the value of the treasure after 10 years is \( \boldsymbol{\$1,250,000} \)