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

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

Flashcard 2: 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 3: 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 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: 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 6: 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 7: 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 8: State the first term in the geometric series $2, 6, 18, ...$.

Answer: $a = 2$. First value in the sequence.

Flashcard 9: What is the common ratio for the series $5, -10, 20, ...$?

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

Flashcard 10: Calculate the sum of the infinite series $1 + 0.5 + 0.25 + ...$.

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

Flashcard 11: 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 12: 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 13: For $a = 5$ and $r = \frac{1}{3}$, find the sum of the infinite series.

Answer: $S = \frac{15}{2}$. Using infinite sum formula: $\frac{5}{1-\frac{1}{3}} = \frac{15}{2}$.

Flashcard 14: Identify the first term in the series $7, 21, 63, ...$.

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

Flashcard 15: 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 16: 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 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: For the series $10, 30, 90, ...$, what is the common ratio?

Answer: $r = 3$. Ratio between consecutive terms is constant at 3.

Flashcard 20: Calculate the sum of the series $1 - 3 + 9 - ...$ for 3 terms.

Answer: $S_3 = 7$. Three terms: $1+(-3)+9 = 7$.

Flashcard 21: 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 22: What is the sum of the first 3 terms of the series $4, 12, 36, ...$?

Answer: $S_3 = 52$. Sum of first three terms: $4+12+36$.

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

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

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

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

Flashcard 29: 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 30: Identify the first term in the series $7, 21, 63, ...$.

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

Flashcard 31: 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 32: 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 33: State the first term in the geometric series $2, 6, 18, ...$.

Answer: $a = 2$. First value in the sequence.

Flashcard 34: Determine the common ratio for the series $10, 20, 40, ...$.

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

Flashcard 35: 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 36: Calculate the sum of the series $1 - 3 + 9 - ...$ for 3 terms.

Answer: $S_3 = 7$. Three terms: $1+(-3)+9 = 7$.

Flashcard 37: 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 38: 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 39: Identify the common ratio for the series $8, -4, 2, ...$.

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

Flashcard 40: 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 41: 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 42: 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 43: Find the common ratio for the series $-2, 4, -8, ...$.

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

Flashcard 44: 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 45: 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 46: 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 47: For the series $10, 30, 90, ...$, what is the common ratio?

Answer: $r = 3$. Ratio between consecutive terms is constant at 3.

Flashcard 48: 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 49: 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 50: What is the sum of the first 3 terms of the series $4, 12, 36, ...$?

Answer: $S_3 = 52$. Sum of first three terms: $4+12+36$.

Flashcard 51: Calculate the sum for the series $1 - 2 + 4 - ...$ for 3 terms.

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

Flashcard 52: Identify the common ratio for the series $8, -4, 2, ...$.

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

Flashcard 53: What is the sum of the series $3, 9, 27, ...$ up to 3 terms?

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

Flashcard 54: 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 55: 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 56: 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 57: Find the common ratio for the series $-2, 4, -8, ...$.

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

Flashcard 58: 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 59: What is the sum of the series $3, 9, 27, ...$ up to 3 terms?

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

Flashcard 60: Identify the first term in the geometric series $3, 6, 12, 24, ...$.

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

Flashcard 61: 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 62: For $a = 5$ and $r = \frac{1}{3}$, find the sum of the infinite series.

Answer: $S = \frac{15}{2}$. Using infinite sum formula: $\frac{5}{1-\frac{1}{3}} = \frac{15}{2}$.

Flashcard 63: 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 64: 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 65: 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 66: 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 67: 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 68: Calculate the sum for the series $1 - 2 + 4 - ...$ for 3 terms.

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

Flashcard 69: Identify the first term in the geometric series $3, 6, 12, 24, ...$.

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

Flashcard 70: Determine the common ratio for the series $10, 20, 40, ...$.

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