Exam 7: Sequences, Series, and Limits

Express 31.14 65 65 65 as a fraction; here the digits 65 keep repeating forever.

Define a recursive sequence using the equations a₁ = 7, . Find the smallest value of n such that a_n = 10.

Evaluate the geometric series. Leave the answer in terms of unsimplified powers of the common ratio.

Which of the following geometric sequences has fourth term a⁵?

Evaluate the geometric series. Leave the answer in terms of unsimplified powers of the common ratio.

Evaluate . Give the exact answer.

is correct?

Express as a fraction.

Evaluate the geometric series. 1 + 5 + 25 + ... +

If and are terms of an arithmetic sequence, find the difference between consecutive terms of this sequence.

Evaluate the arithmetic series.

Find the sum of all three-digit positive integers.

Express as a fraction.

Find two numbers between 6 and 18 such that they form an arithmetic sequence with 6 and 18.

Find the 5th row of the Pascal's Triangle.

Evaluate the geometric series.

Consider the fable where one grain of rice is placed on the first square of a chessboard, then two grains on the square, then four grains on the third square, and so on, doubling the number of grains placed on each square. Find the total number of grains of rice on the first 15 squares of the chessboard.

Find the fifth term of the recursive sequence a₁ = a, a₂ = b, a_{n + 2} = a_n*a_{n + 1} for n $\ge$ 1.

Find the first four terms of the recursive sequence a₁ = 3, a_{n + 1} = 3a_n - 2 for n ≥ 1.

Evaluate