Question

the table shows the
year | value $
1 | 8,450
2 | 9,200
3 | 9,950
4 | 10,700
select all the true statements about the growth of the total investment value.
□ the growth is linear.
□ the growth is exponential.
□ the growth is neither linear nor exponential.
□ the principal is 8,450 - 750.
□ the principal is (0.6) 8,450.

Explanation:

Step1: Check for linear growth

Calculate the differences between consecutive values:

• From Year 1 to Year 2: \( 9200 - 8450 = 750 \)
• From Year 2 to Year 3: \( 9950 - 9200 = 750 \)
• From Year 3 to Year 4: \( 10700 - 9950 = 750 \)

Since the difference between consecutive years is constant (\( 750 \)), the growth is linear.

Step2: Analyze principal statements

• "The principal is \( 8450 - 750 \)": If the growth is linear with a constant difference of \( 750 \), the principal (initial value) is Year 1 value minus the growth? Wait, no. Wait, Year 1 value is \( 8450 \). If we assume the growth starts from a principal, and the first year's value is principal + growth? Wait, no, in linear growth, the formula is \( V(n) = V(1) + (n - 1)d \), where \( d = 750 \). So \( V(1) = 8450 \), which would be principal (if we consider Year 1 as the first period). But the statement says "The principal is \( 8450 - 750 \)". Let's check: \( 8450 - 750 = 7700 \). Is that the principal? Wait, maybe the problem is considering the principal as the value before Year 1. If Year 1 value is principal + 750, then principal would be \( 8450 - 750 = 7700 \). But let's check the other statement: "The principal is \( (0.6)8450 \)". \( 0.6 \times 8450 = 5070 \), which is not related. So the first principal statement: if we consider that Year 1's value is principal + 750 (growth in Year 1), then principal is \( 8450 - 750 \). Wait, but in linear growth, the formula is \( V(n) = P + nd \), where \( P \) is principal, \( d \) is annual growth. Wait, no, for \( n = 1 \), \( V(1) = P + d \), \( n = 2 \), \( V(2) = P + 2d \), etc. Wait, in our case, \( V(1) = 8450 \), \( V(2) = 9200 = 8450 + 750 \), \( V(3) = 9200 + 750 = 8450 + 2 \times 750 \), \( V(4) = 8450 + 3 \times 750 \). So if we write \( V(n) = P + (n - 1)d \), then for \( n = 1 \), \( V(1) = P \), so \( P = 8450 \). But the statement says "The principal is \( 8450 - 750 \)". Maybe the problem is considering the principal as the value before Year 1, so Year 0 value. Then \( V(1) = V(0) + 750 \), so \( V(0) = 8450 - 750 = 7700 \). So that statement could be considered true if we take principal as Year 0 value. The other principal statement: \( 0.6 \times 8450 = 5070 \), which is not related, so that's false.

Now, the growth is linear (since constant difference), so "The growth is linear" is true. "The growth is exponential" is false (exponential would have constant ratio, but ratios are \( 9200/8450 \approx 1.0887 \), \( 9950/9200 \approx 1.0815 \), not constant). "The growth is neither linear nor exponential" is false. "The principal is \( 8450 - 750 \)" – if we consider principal as Year 0 value, then yes (since Year 1 is Year 0 + 750). "The principal is \( (0.6)8450 \)" is false.

Wait, but let's re-examine the principal statements. Maybe the problem has a typo, but let's go with the calculations.

Answer:

• The growth is linear.
• The principal is \( 8,450 - 750 \).