B.Sc Mathematics Sequences, Probability and Counting Theory — practice questions — page 2

1. In slope-intercept form the slope is the common difference, so the coefficient of n is -50. 2. The vertical intercept is found by subtracting the common difference from the first term. 3. That gives 200 - (-50) = 200 + 50 = 250. 4. So the formula is a_n = -50n + 250. 5. Checking n = 1 gives -50 + 250 = 200, which is the first term, so the formula is right. 6. Using 200 as the intercept would put the first term at 150, which is the trap in that option. 7. Adding rather than subtracting the common difference gives 150, and flipping the sign of the slope reverses the whole sequence. _Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 13 "Sequences, Probability and Counting Theory", section 13.2 Arithmetic Sequences_

1. Subtract the first term from the second: 12 - 2 = 10, so d = 10. 2. Substitute the first term and d into a_n = a_1 + d(n - 1), giving a_n = 2 + 10(n - 1). 3. Expanding gives a_n = 2 + 10n - 10, which simplifies to a_n = 10n - 8. 4. Testing n = 1 gives 10 - 8 = 2, the first term, and n = 3 gives 30 - 8 = 22. 5. Forgetting to expand the bracket leaves a_n = 10n + 2, which puts 12 at the first position. _Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 13 "Sequences, Probability and Counting Theory", section 13.2 Arithmetic Sequences_

1. Find the common difference: 1 - 8 = -7. 2. Substitute into a_n = a_1 + d(n - 1) to get a_n = 8 + (-7)(n - 1). 3. Expanding gives a_n = 8 - 7n + 7, which simplifies to a_n = 15 - 7n. 4. Set the last term equal to this: -41 = 15 - 7n. 5. Subtracting 15 gives -56 = -7n, so n = 8. 6. There are eight terms in the sequence. 7. Counting how many steps of -7 fit between 8 and -41 gives 7, which is one fewer than the number of terms, and that off-by-one is the trap. _Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 13 "Sequences, Probability and Counting Theory", section 13.2 Arithmetic Sequences_