Lesson: Chapter - 12

Explanations

1.C

If a statement is false, the contrapositive of the statement will also be false. To find the contrapositive, you need to take the opposite of both parts of the statement and then switch the order. The contrapositive of “If it rains, it pours,” is “If it doesn’t pour, it doesn’t rain.”

2.B

The formula for the n^th term of an arithmetic sequence is a_n = a₁ + (n – 1)d, where d is the difference between the terms of an arithmetic sequence.

If the first term of a sequence is –3, and d = –3, then a_n = –3 –3n + 3 = –3n. So, the 30^th term is –3(30) = –90.

3.D

This sequence is an arithmetic sequence since the difference between each term is constant. The formula for the sum of the first n terms of an arithmetic sequence is:

To use this formula for this question, first calculate the values of a₁ and a₁₀₀ by plugging n = 1 and n = 100 into the given formula a_n = 6n – 3. So, we find that a₁ = 6 – 3 = 3 and a₁₀₀ = 600 – 3 = 597. The sum of the first 100 terms is therefore:

4.D

To answer this question quickly and efficiently, you need to know the formula for the sum of the first n terms of a geometric sequence:

where r is the common ratio of the sequence. In this problem b₁ = ³/₂(2¹) = 3 and r = 2, so the formula yields the answer 3069.

5.A

This question throws a little curve-ball at you because function is undefined at x = –2, since –2² – 4 = 0. However, the denominator can be factored into (x – 2)(x + 2). Then (x + 2) can be canceled from the numerator and denominator, leaving ¹/_x–2 as the function. Evaluating this function at x = –2, you see the limit is –¹/₄.

6.D

The powers of i repeat themselves in a cycle of four, that is iⁿ = iⁿ⁺⁴. Since i⁴ = 1, i⁵ must equal i. You can also reduce i¹⁴ by noticing that it equals i¹² × i². Since 12 is a multiple of 4, i¹² equals 1, so

So 3(i – i¹⁴) = 3(i – (–1)) = 3(i + 1) = 3i + 3.

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