Working with Geometric Series - AP Calculus BC

1. State the formula for the $n$-th term of a geometric series.

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

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

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

3. What is the first term $a$ if the series is $a, ar, ar^2, ...$?

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

4. State the first term in the geometric series $2, 6, 18, ...$.

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

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

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

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

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

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

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

8. Identify the first term in the series $7, 21, 63, ...$.

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

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

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

10. State the condition for a geometric series to be infinite and divergent.

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

11. Find the sum of the infinite series $2 + 1 + 0.5 + + ...$.

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

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

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

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

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

14. For the series $5, 15, 45, ...$, what is the common ratio?

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

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

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

16. What is the common ratio in a geometric series?

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

17. Calculate the sum of the series $1 - 3 + 9 - ...$ for 3 terms.

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

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

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

19. Identify the common ratio for the series $8, -4, 2, ...$.

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

20. State the formula for the sum of an infinite geometric series.

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

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

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

22. State the formula to find the $n$-th term in a geometric sequence.

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

23. What condition must the common ratio meet for an infinite geometric series to converge?

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

24. For the series $10, 30, 90, ...$, what is the common ratio?

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

25. What is the condition for a geometric series to be finite?

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

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

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

27. What is the sum of the first 3 terms of the series $4, 12, 36, ...$?

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

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

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

29. Identify the condition for a geometric sequence to be finite.

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

30. What formula is used to determine the sum of a convergent infinite series?

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

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

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

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

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

33. Calculate the sum for the series $1 - 2 + 4 - ...$ for 3 terms.

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

34. Identify the first term in the geometric series $3, 6, 12, 24, ...$.

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

35. Determine the common ratio for the series $10, 20, 40, ...$.

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

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