Given the first term $$a_1 = 3$$ and common ratio $$r = 5$$ , find the 7th term of the sequence.

The 7th term is calculated as follows: $$a_7 = 3 \times 5^{(7-1)} = 3 \times 15625 = 46875$$ .

Evaluate the geometric series: $$3 + 15 + 75 + 375 + ...$$ for $$n = 6$$ .

The sum is $$S_6 = 3 \times \frac{1 - 5^6}{1 - 5} = 3 \times \frac{1 - 15625}{-4} = 3 \times 3906 = 11718$$ .

What is the common ratio of the geometric sequence -1, 6, -36, 216, ...?

The common ratio is $$r = \frac{6}{-1} = -6$$ .

If $$a_1 = 3$$ and $$r = -2$$ , what is the value of the 14th term?

The 14th term is calculated as: $$a_{14} = 3 \times (-2)^{(14-1)} = 3 \times (-2)^{13} = -24576$$ .

What is the relationship between the common ratio and the terms of a geometric sequence?

The common ratio determines how each term in the sequence is related to the previous term; it is the factor by which we multiply to get from one term to the next.

How do you identify a geometric sequence from a list of numbers?

A sequence is geometric if the ratio of consecutive terms is constant.

What happens to the sum of a geometric series if the common ratio is between -1 and 1?

If the common ratio is between -1 and 1, the series converges to a finite sum.

What is the sum of an infinite geometric series with first term $$a_1$$ and common ratio $$r$$ (where |r|

The sum is given by: $$S = \frac{a_1}{1 - r}$$ .

If a geometric sequence has a common ratio of 0, what can be said about the sequence?

If the common ratio is 0, all terms after the first term will be 0, resulting in a sequence that is constant at the first term.

Explain how to derive the formula for the sum of a finite geometric series.

To derive the formula, multiply the sum by the common ratio, subtract from the original sum, and simplify to isolate the sum.

Geometric Sequences/Series Flashcards

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15 questions

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1.

FLASHCARD QUESTION

Front

Define a geometric sequence.

Back

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

2.

FLASHCARD QUESTION

Front

What is the formula for the nth term of a geometric sequence?

Back

3.

FLASHCARD QUESTION

Front

Back

4.

FLASHCARD QUESTION

Front

What is a geometric series?

Back

A geometric series is the sum of the terms of a geometric sequence. It can be finite or infinite.

5.

FLASHCARD QUESTION

Front

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

Back

6.

FLASHCARD QUESTION

Front

Back

7.

FLASHCARD QUESTION

Front

Find the common ratio of the sequence: 64, -16, 4, -1, ...

Back

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