Front

What is an arithmetic sequence?

Back

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference (d).

Front

How do you find the common difference (d) in an arithmetic sequence?

Back

To find the common difference (d), subtract any term from the subsequent term: d = a_{n+1} - a_n.

Front

If a_{19} = 102 and a_{36} = 221, how do you find d?

Back

Use the formula: d = (a_{36} - a_{19}) / (36 - 19). Thus, d = (221 - 102) / 17 = 7.

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What is the formula for the nth term of an arithmetic sequence?

Back

The nth term (a_n) can be calculated using the formula: a_n = a_1 + (n-1)d, where a_1 is the first term and d is the common difference.

Front

How do you calculate the sum of the first n terms of an arithmetic sequence?

Back

The sum (S_n) of the first n terms can be calculated using the formula: S_n = n/2 * (a_1 + a_n).

Front

What is the sum of the first 100 positive multiples of 3?

Back

The sum is 15,150, calculated using the formula for the sum of an arithmetic series.

Front

If a_1 = 5 and d = 3, how do you find a_6?

Back

Use the formula: a_6 = a_1 + (6-1)d = 5 + 5*3 = 20.

Front

What is the total vertical distance traveled by a ball that rebounds 95% of the distance it falls?

Back

The total distance can be calculated as an infinite geometric series: Total Distance = 12 + 2 * (12 * 0.95 + 12 * 0.95^2 + ...). The answer is 480.

Front

What is a geometric series?

Back

A geometric series is the sum of the terms of a geometric sequence, where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

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What is the formula for the sum of the first n terms of a geometric series?

Back

The sum (S_n) of the first n terms of a geometric series can be calculated using: S_n = a(1 - r^n) / (1 - r), where a is the first term and r is the common ratio.