Free Pre-Calculus Sequences and Series Questions and Answers

Question 1

Create the geometric sequence's formula: -2, 10, -50, 250...

Accepted Answer

aₙ = -2(-5)^n-1

Answer

A geometric sequence has a first term (a₁) and a common ratio (r), with the formula aₙ = a₁ * r^(n-1). From the sequence -2, 10, -50, 250..., the first term a₁ is -2. The common ratio r is found by dividing any term by its preceding term: 10 / (-2) = -5. Substituting these values into the formula gives aₙ = -2(-5)^(n-1).

Question 2

Eliya's Bakery has recently launched and is expanding its bread products. For instance, the bakery produced 150 carrot cakes in January, 133 in February, 116 in March, and 99 in April. What type of sequence is this?

Accepted Answer

Arithmetic

Answer

To classify the sequence, we check for a common difference (arithmetic) or a common ratio (geometric). The differences between consecutive terms are: 133 - 150 = -17, 116 - 133 = -17, and 99 - 116 = -17. Since there is a constant difference of -17, the sequence is arithmetic. It is not geometric as the ratios between terms are not constant.

Question 3

Determine the sum, if it exists, of the infinite geometric series: 200 - 100 + 50 - 25 +

Accepted Answer

400/ 3

Answer

This is an infinite geometric series with a first term (a) of 200 and a common ratio (r) of -100/200 = -1/2. Since the absolute value of the common ratio (|r| = 1/2) is less than 1, the sum exists. Using the formula S = a / (1 - r), we get S = 200 / (1 - (-1/2)) = 200 / (3/2) = 400/3.

Question 4

A geometric sequence's third term is 96 and its fifth term is 1536. What is the sum of the sequence's first ten terms?

Accepted Answer

2,097,150

Answer

First, find the common ratio (r) using the terms a_5 = a_3 * r^2, which gives 1536 = 96 * r^2, so r^2 = 16, and r = 4 (since terms are positive). Then, find the first term (a_1) using a_3 = a_1 * r^2, so 96 = a_1 * 4^2, which means a_1 = 6. Finally, use the sum formula for a geometric series S_n = a_1 * (1 - r^n) / (1 - r) to find S_10 = 6 * (1 - 4^10) / (1 - 4) = 2,097,150.

Question 5

Determine the common difference 97, 86, 75, 64, ...

Accepted Answer

-11

Answer

The common difference in an arithmetic sequence is found by subtracting any term from its succeeding term. For this sequence, 86 - 97 = -11. We can verify this by checking other consecutive terms, such as 75 - 86 = -11 and 64 - 75 = -11, confirming the common difference is -11.

PRE CALCULUS Practice Test

PRE CALCULUS Practice Test

Free Pre-Calculus Sequences and Series Questions and Answers