In the sequence shown, b1 = 1 and for all integers n >= 2, bn = 2bn-1 + r, where r is a positive integer. The sum of b1, b2, and b3 is 35. Quantity A: r, Quantity B: 7

Rationale

The two quantities are equal.

Given the recurrence relation \( b_n = 2b_{n-1} + r \) starting with \( b_1 = 1 \), we can compute \( b_2 \) and \( b_3 \). The sum \( b_1 + b_2 + b_3 = 35 \) allows us to solve for \( r \). By calculating, we find that \( r = 7 \), leading to the conclusion that Quantity A and Quantity B are equal.

A (Quantity A is greater.) YOUR ANSWER

If Quantity A were greater than Quantity B, then \( r \) would need to exceed 7. However, our calculations show that \( r \) equals 7, which contradicts this option.

B (Quantity B is greater.) YOUR ANSWER

This option suggests that \( r \) is less than 7. Since we determined \( r = 7 \), this choice cannot be valid because it would imply \( r \) is a smaller integer, which is incorrect.

C (The two quantities are equal.) CORRECT

This is correct because our calculations reveal that \( r = 7 \), making both Quantity A and Quantity B equal. The relationship holds true based on the given conditions and calculations.

D (The relationship cannot be determined from the information given.) YOUR ANSWER

This option would imply ambiguity in determining \( r \). However, since we have a clear equation and specific values that sum to 35, we can definitively determine \( r \) to be 7.

Conclusion

The problem clearly defines a sequence governed by a recurrence relation, which allows us to calculate \( r \). After solving for \( r \) using the sum of the first three terms, we confirm that \( r = 7 \). Thus, both quantities are equal, affirming our answer.

In the sequence shown, b1 = 1 and for all integers n >= 2, bn = 2bn-1 + r, where r is a positive integer. The sum of b1, b2, and b3 is 35. Quantity A: r, Quantity B: 7

The two quantities are equal.

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