Working with Geometric Series - AP Calculus BC
Card 1 of 30
If $a = 2$ and $r = \frac{1}{2}$, what is the sum of the infinite series?
If $a = 2$ and $r = \frac{1}{2}$, what is the sum of the infinite series?
Tap to reveal answer
$S = 4$. Applying the infinite sum formula directly.
$S = 4$. Applying the infinite sum formula directly.
← Didn't Know|Knew It →
Find the common ratio for the series $-2, 4, -8, ...$.
Find the common ratio for the series $-2, 4, -8, ...$.
Tap to reveal answer
$r = -2$. Each term multiplies the previous by $-2$.
$r = -2$. Each term multiplies the previous by $-2$.
← Didn't Know|Knew It →
Determine the common ratio for the series $10, 20, 40, ...$.
Determine the common ratio for the series $10, 20, 40, ...$.
Tap to reveal answer
$r = 2$. Each term doubles the previous one.
$r = 2$. Each term doubles the previous one.
← Didn't Know|Knew It →
What is the formula for the sum of the first $n$ terms of a geometric series?
What is the formula for the sum of the first $n$ terms of a geometric series?
Tap to reveal answer
$S_n = a \frac{1-r^n}{1-r}$, $r \neq 1$. For finite geometric series when $r \neq 1$.
$S_n = a \frac{1-r^n}{1-r}$, $r \neq 1$. For finite geometric series when $r \neq 1$.
← Didn't Know|Knew It →
State the formula for the sum of an infinite geometric series.
State the formula for the sum of an infinite geometric series.
Tap to reveal answer
$S = \frac{a}{1-r}$, where $|r| < 1$. Valid only when the series converges.
$S = \frac{a}{1-r}$, where $|r| < 1$. Valid only when the series converges.
← Didn't Know|Knew It →
What is the common ratio in a geometric series?
What is the common ratio in a geometric series?
Tap to reveal answer
The ratio $r$ between consecutive terms. Found by dividing any term by the previous term.
The ratio $r$ between consecutive terms. Found by dividing any term by the previous term.
← Didn't Know|Knew It →
Find the sum of the infinite series $2 + 1 + 0.5 + + ...$.
Find the sum of the infinite series $2 + 1 + 0.5 + + ...$.
Tap to reveal answer
$S = 4$. Using $S = \frac{a}{1-r} = \frac{2}{1-0.5} = 4$.
$S = 4$. Using $S = \frac{a}{1-r} = \frac{2}{1-0.5} = 4$.
← Didn't Know|Knew It →
What condition must the common ratio meet for an infinite geometric series to converge?
What condition must the common ratio meet for an infinite geometric series to converge?
Tap to reveal answer
$|r| < 1$. Ensures the series approaches a finite limit.
$|r| < 1$. Ensures the series approaches a finite limit.
← Didn't Know|Knew It →
State the formula for the $n$-th term of a geometric series.
State the formula for the $n$-th term of a geometric series.
Tap to reveal answer
$a_n = ar^{n-1}$. General term formula for geometric sequences.
$a_n = ar^{n-1}$. General term formula for geometric sequences.
← Didn't Know|Knew It →
For the series $5, 15, 45, ...$, what is the common ratio?
For the series $5, 15, 45, ...$, what is the common ratio?
Tap to reveal answer
$r = 3$. Each term is multiplied by 3 to get the next.
$r = 3$. Each term is multiplied by 3 to get the next.
← Didn't Know|Knew It →
What formula is used to determine the sum of a convergent infinite series?
What formula is used to determine the sum of a convergent infinite series?
Tap to reveal answer
$S = \frac{a}{1-r}$ for $|r| < 1$. Standard convergent infinite series formula.
$S = \frac{a}{1-r}$ for $|r| < 1$. Standard convergent infinite series formula.
← Didn't Know|Knew It →
Calculate the sum for the series $1 - 2 + 4 - ...$ for 3 terms.
Calculate the sum for the series $1 - 2 + 4 - ...$ for 3 terms.
Tap to reveal answer
$S_3 = 3$. Three terms: $1+(-2)+4 = 3$.
$S_3 = 3$. Three terms: $1+(-2)+4 = 3$.
← Didn't Know|Knew It →
Identify the first term in the series $7, 21, 63, ...$.
Identify the first term in the series $7, 21, 63, ...$.
Tap to reveal answer
$a = 7$. The starting term of the sequence.
$a = 7$. The starting term of the sequence.
← Didn't Know|Knew It →
State the formula to find the $n$-th term in a geometric sequence.
State the formula to find the $n$-th term in a geometric sequence.
Tap to reveal answer
$a_n = ar^{n-1}$. Standard formula for geometric sequence terms.
$a_n = ar^{n-1}$. Standard formula for geometric sequence terms.
← Didn't Know|Knew It →
Calculate the sum of the infinite series $1 + 0.5 + 0.25 + ...$.
Calculate the sum of the infinite series $1 + 0.5 + 0.25 + ...$.
Tap to reveal answer
$S = 2$. Geometric series with $a=1$, $r=0.5$.
$S = 2$. Geometric series with $a=1$, $r=0.5$.
← Didn't Know|Knew It →
Identify the common ratio for the series $8, -4, 2, ...$.
Identify the common ratio for the series $8, -4, 2, ...$.
Tap to reveal answer
$r = -\frac{1}{2}$. Alternating pattern with factor of $-\frac{1}{2}$.
$r = -\frac{1}{2}$. Alternating pattern with factor of $-\frac{1}{2}$.
← Didn't Know|Knew It →
State the condition for a geometric series to be infinite and divergent.
State the condition for a geometric series to be infinite and divergent.
Tap to reveal answer
$|r| \geq 1$. When the absolute value equals or exceeds 1.
$|r| \geq 1$. When the absolute value equals or exceeds 1.
← Didn't Know|Knew It →
What is the first term $a$ if the series is $a, ar, ar^2, ...$?
What is the first term $a$ if the series is $a, ar, ar^2, ...$?
Tap to reveal answer
$a$ is the initial term of the series. By definition in the geometric sequence notation.
$a$ is the initial term of the series. By definition in the geometric sequence notation.
← Didn't Know|Knew It →
What is the common ratio for the series $5, -10, 20, ...$?
What is the common ratio for the series $5, -10, 20, ...$?
Tap to reveal answer
$r = -2$. Pattern shows multiplication by $-2$ each time.
$r = -2$. Pattern shows multiplication by $-2$ each time.
← Didn't Know|Knew It →
What is the sum of the series $3, 9, 27, ...$ up to 3 terms?
What is the sum of the series $3, 9, 27, ...$ up to 3 terms?
Tap to reveal answer
$S_3 = 39$. Sum: $3+9+27 = 39$.
$S_3 = 39$. Sum: $3+9+27 = 39$.
← Didn't Know|Knew It →
What is the condition for a geometric series to be finite?
What is the condition for a geometric series to be finite?
Tap to reveal answer
A finite number of terms. Has a specified endpoint, unlike infinite series.
A finite number of terms. Has a specified endpoint, unlike infinite series.
← Didn't Know|Knew It →
Calculate the sum of the infinite series $3 - 1 + \frac{1}{3} - \frac{1}{9} + ...$.
Calculate the sum of the infinite series $3 - 1 + \frac{1}{3} - \frac{1}{9} + ...$.
Tap to reveal answer
$S = \frac{9}{4}$. Alternating series with $a=3$, $r=-\frac{1}{3}$.
$S = \frac{9}{4}$. Alternating series with $a=3$, $r=-\frac{1}{3}$.
← Didn't Know|Knew It →
Identify the condition for a geometric sequence to be finite.
Identify the condition for a geometric sequence to be finite.
Tap to reveal answer
The number of terms is finite. Distinguished from infinite series by term count.
The number of terms is finite. Distinguished from infinite series by term count.
← Didn't Know|Knew It →
Find the sum of the infinite series $3, -1, \frac{1}{3}, ...$ if $|r| < 1$.
Find the sum of the infinite series $3, -1, \frac{1}{3}, ...$ if $|r| < 1$.
Tap to reveal answer
$S = \frac{9}{4}$. Converges since $|r| = \frac{1}{3} < 1$.
$S = \frac{9}{4}$. Converges since $|r| = \frac{1}{3} < 1$.
← Didn't Know|Knew It →
What is the sum of the series $1 - 3 + 9 - ...$ for the first 3 terms?
What is the sum of the series $1 - 3 + 9 - ...$ for the first 3 terms?
Tap to reveal answer
$S_3 = 7$. Alternating signs with $a=1$, $r=-3$.
$S_3 = 7$. Alternating signs with $a=1$, $r=-3$.
← Didn't Know|Knew It →
Determine the sum of the series $2 + 4 + 8 + ...$ up to 4 terms.
Determine the sum of the series $2 + 4 + 8 + ...$ up to 4 terms.
Tap to reveal answer
$S_4 = 30$. Sum: $2+4+8+16 = 30$.
$S_4 = 30$. Sum: $2+4+8+16 = 30$.
← Didn't Know|Knew It →
For $a = 5$ and $r = -\frac{1}{3}$, find the sum of the infinite series.
For $a = 5$ and $r = -\frac{1}{3}$, find the sum of the infinite series.
Tap to reveal answer
$S = \frac{15}{4}$. Using $S = \frac{a}{1-r}$ with given values.
$S = \frac{15}{4}$. Using $S = \frac{a}{1-r}$ with given values.
← Didn't Know|Knew It →
What is the sum of the series $1 + 2 + 4 + 8 + ...$ up to 4 terms?
What is the sum of the series $1 + 2 + 4 + 8 + ...$ up to 4 terms?
Tap to reveal answer
$S_4 = 15$. Sum: $1+2+4+8 = 15$.
$S_4 = 15$. Sum: $1+2+4+8 = 15$.
← Didn't Know|Knew It →
What is the sum of the infinite series $1 + \frac{1}{2} + \frac{1}{4} + ...$?
What is the sum of the infinite series $1 + \frac{1}{2} + \frac{1}{4} + ...$?
Tap to reveal answer
$S = 2$. Series with $a=1$, $r=\frac{1}{2}$ converges.
$S = 2$. Series with $a=1$, $r=\frac{1}{2}$ converges.
← Didn't Know|Knew It →
Identify the first term in the geometric series $3, 6, 12, 24, ...$.
Identify the first term in the geometric series $3, 6, 12, 24, ...$.
Tap to reveal answer
$a = 3$. The initial value before any multiplication by $r$.
$a = 3$. The initial value before any multiplication by $r$.
← Didn't Know|Knew It →