AP Calculus BC Flashcards: Working With Geometric Series

What this deck covers

This deck focuses on Working With Geometric Series, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: If $a = 2$ and $r = \frac{1}{2}$, what is the sum of the infinite series?

Answer: $S = 4$. Applying the infinite sum formula directly.

Flashcard 2: Find the common ratio for the series $-2, 4, -8, ...$.

Answer: $r = -2$. Each term multiplies the previous by $-2$.

Flashcard 3: Determine the common ratio for the series $10, 20, 40, ...$.

Answer: $r = 2$. Each term doubles the previous one.

Flashcard 4: What is the formula for the sum of the first $n$ terms of a geometric series?

Answer: $S_n = a \frac{1-r^n}{1-r}$, $r \neq 1$. For finite geometric series when $r \neq 1$.

Flashcard 5: State the formula for the sum of an infinite geometric series.

Answer: $S = \frac{a}{1-r}$, where $|r| < 1$. Valid only when the series converges.

Flashcard 6: What is the common ratio in a geometric series?

Answer: The ratio $r$ between consecutive terms. Found by dividing any term by the previous term.

Flashcard 7: Find the sum of the infinite series 2 + 1 + 0.5 +  + ....

Answer: $S = 4$. Using $S = \frac{a}{1-r} = \frac{2}{1-0.5} = 4$.

Flashcard 8: What condition must the common ratio meet for an infinite geometric series to converge?

Answer: $|r| < 1$. Ensures the series approaches a finite limit.

Flashcard 9: State the formula for the $n$-th term of a geometric series.

Answer: $a_n = ar^{n-1}$. General term formula for geometric sequences.

Flashcard 10: For the series $5, 15, 45, ...$, what is the common ratio?

Answer: $r = 3$. Each term is multiplied by 3 to get the next.

Flashcard 11: What formula is used to determine the sum of a convergent infinite series?

Answer: $S = \frac{a}{1-r}$ for $|r| < 1$. Standard convergent infinite series formula.

Flashcard 12: Calculate the sum for the series $1 - 2 + 4 - ...$ for 3 terms.

Answer: $S_3 = 3$. Three terms: $1+(-2)+4 = 3$.

Flashcard 13: Identify the first term in the series $7, 21, 63, ...$.

Answer: $a = 7$. The starting term of the sequence.

Flashcard 14: State the formula to find the $n$-th term in a geometric sequence.

Answer: $a_n = ar^{n-1}$. Standard formula for geometric sequence terms.

Flashcard 15: Calculate the sum of the infinite series $1 + 0.5 + 0.25 + ...$.

Answer: $S = 2$. Geometric series with $a=1$, $r=0.5$.

Flashcard 16: Identify the common ratio for the series $8, -4, 2, ...$.

Answer: $r = -\frac{1}{2}$. Alternating pattern with factor of $-\frac{1}{2}$.

Flashcard 17: State the condition for a geometric series to be infinite and divergent.

Answer: $|r| \geq 1$. When the absolute value equals or exceeds 1.

Flashcard 18: What is the first term $a$ if the series is $a, ar, ar^2, ...$?

Answer: $a$ is the initial term of the series. By definition in the geometric sequence notation.

Flashcard 19: What is the common ratio for the series $5, -10, 20, ...$?

Answer: $r = -2$. Pattern shows multiplication by $-2$ each time.

Flashcard 20: What is the sum of the series $3, 9, 27, ...$ up to 3 terms?

Answer: $S_3 = 39$. Sum: $3+9+27 = 39$.

Flashcard 21: What is the condition for a geometric series to be finite?

Answer: A finite number of terms. Has a specified endpoint, unlike infinite series.

Flashcard 22: Calculate the sum of the infinite series $3 - 1 + \frac{1}{3} - \frac{1}{9} + ...$.

Answer: $S = \frac{9}{4}$. Alternating series with $a=3$, $r=-\frac{1}{3}$.

Flashcard 23: Identify the condition for a geometric sequence to be finite.

Answer: The number of terms is finite. Distinguished from infinite series by term count.

Flashcard 24: Find the sum of the infinite series $3, -1, \frac{1}{3}, ...$ if $|r| < 1$.

Answer: $S = \frac{9}{4}$. Converges since $|r| = \frac{1}{3} < 1$.

Flashcard 25: What is the sum of the series $1 - 3 + 9 - ...$ for the first 3 terms?

Answer: $S_3 = 7$. Alternating signs with $a=1$, $r=-3$.

Flashcard 26: Determine the sum of the series $2 + 4 + 8 + ...$ up to 4 terms.

Answer: $S_4 = 30$. Sum: $2+4+8+16 = 30$.

Flashcard 27: For $a = 5$ and $r = -\frac{1}{3}$, find the sum of the infinite series.

Answer: $S = \frac{15}{4}$. Using $S = \frac{a}{1-r}$ with given values.

Flashcard 28: What is the sum of the series $1 + 2 + 4 + 8 + ...$ up to 4 terms?

Answer: $S_4 = 15$. Sum: $1+2+4+8 = 15$.

Flashcard 29: What is the sum of the infinite series $1 + \frac{1}{2} + \frac{1}{4} + ...$?

Answer: $S = 2$. Series with $a=1$, $r=\frac{1}{2}$ converges.

Flashcard 30: Identify the first term in the geometric series $3, 6, 12, 24, ...$.

Answer: $a = 3$. The initial value before any multiplication by $r$.