This question tests your understanding of geometric series—the sum of terms from a geometric sequence—and how to derive and use the formula $S_n = a \frac{1 - r^n}{1 - r}$ to calculate these sums efficiently. A geometric series is the sum of terms from a geometric sequence: here we have 2, 2(1.1), 2(1.1) squared, ..., 2(1.1) to the 9, where each term is multiplied by 1.1 to get the next. Rather than computing each power of 1.1 and adding, we use the formula $S_n = a \frac{1 - r^n}{1 - r}$. To identify parameters: a = 2 (first term), r = 1.1 (common ratio), and n = 10 (count carefully: from 2(1.1) to the 0 up to 2(1.1) to the 9 gives us 10 terms total—don't forget that (1.1) to the 0 = 1!). The formula gives $S_{10} = 2 \frac{1 - (1.1)^{10}}{1 - 1.1}$. Choice B correctly shows this expression with n = 10. Choice A incorrectly uses n = 9, missing that we start with 2 = 2(1.1) to the 0 and end with 2(1.1) to the 9, giving 10 terms total. When the last term has exponent 9, there are 10 terms because we include the 0th power! The three-step geometric series strategy: (1) Count terms by looking at exponents: from $r^0$ to $r^9$ is 10 terms. (2) Identify a as the coefficient (here 2). (3) Apply the formula carefully. Remember: when terms go from $r^0$ to $r^k$, you have (k+1) terms total!

This question tests your understanding of geometric series—identifying parameters and applying the formula S_n = a(1 - $r^n$)/(1 - r) to calculate sums efficiently. A geometric series 5 + 15 + 45 + 135 + 405 has first term a = 5, and checking the ratio: 15/5 = 3, 45/15 = 3, so common ratio r = 3. Counting the terms: we have 5 terms total, so n = 5. Using the formula: S_5 = 5(1 - $3^5$)/(1 - 3) = 5(1 - 243)/(−2) = 5(−242)/(−2) = 5(121) = 605. Choice A correctly identifies a = 5, r = 3, n = 5 and calculates S_5 = 605. Choice B incorrectly states a = 15 (that's the second term, not first!), while C has wrong values for both r and n, and D miscounts n = 4 instead of 5. The three-step strategy: (1) Identify first term a (what's written first), (2) Find r by dividing any term by the previous one, (3) Count the terms carefully for n. Remember: the first term is a, not ar!

This question tests your understanding of geometric series—the sum of terms from a geometric sequence—and identifying parameters to apply S_n = a(1 - $r^n$)/(1 - r). A geometric series is the sum of terms from a geometric sequence: here 5 + 15 + 45 + 135 + 405 (multiply by 3 each time, 5 terms). Rather than adding manually, use the formula with a=5, r=3, n=5: S_5 = 5(1 - $3^5$)/(1 - 3) = 5(1 - 243)/(-2) = 5(-242)/(-2) = 605. Verify: 5+15=20, +45=65, +135=200, +405=605—perfect! Choice A correctly identifies a=5, r=3, n=5 and computes S_5=605. Choice C mistakes a=15 (second term) instead of first term 5; always start with the initial term! The three-step strategy: (1) List terms to find a (first), r (ratio), n (count), (2) Check r ≠1, (3) Substitute carefully. Mastering parameter ID builds confidence for any series—keep going!

This question tests your understanding of geometric series—the sum of terms from a geometric sequence—and how to derive and use the formula S sub n = a(1 - r to the n) divided by (1 - r) to calculate these sums efficiently. A geometric series with negative common ratio alternates signs: here with a = 12 and r = -1/3, we get 12, -4, 4/3, -4/9, 4/27, ... Rather than dealing with alternating fractions, we use the formula S sub n = a(1 - r to the n) divided by (1 - r). To find S sub 5: substitute a = 12, r = -1/3, n = 5. First calculate (-1/3) to the 5 = -1/243 (negative because odd power of negative number). Then S sub 5 = 12(1 - (-1/243)) divided by (1 - (-1/3)) = 12(1 + 1/243) divided by (1 + 1/3) = 12(244/243) divided by (4/3) = 12(244/243) × (3/4) = 181/9. Choice A correctly shows this calculation, though it could be simplified further. Choice B incorrectly uses n = 4 instead of n = 5—when finding S sub 5, use 5 terms! Also note that (-1/3) to the 4 = 1/81 (positive for even power), while (-1/3) to the 5 = -1/243 (negative for odd power). The three-step strategy for negative ratios: (1) Calculate r to the n carefully—negative base with odd exponent stays negative. (2) Handle signs in (1 - r to the n) and (1 - r) separately. (3) Simplify the final fraction. Negative ratios create alternating series, but the formula handles them perfectly!

This question tests your understanding of geometric series—the sum of terms from a geometric sequence—and how to derive and use the formula S_n = a(1 - $r^n$)/(1 - r) to calculate these sums efficiently. The key distinction is between finite sums (S_n for specific n) and infinite series convergence. For any finite n, we always use S_n = a(1 - $r^n$)/(1 - r), regardless of whether |r| < 1 or not—the formula works for all r ≠ 1. When |r| < 1 (as with r = 1/4), the infinite series converges to a/(1 - r), but for any specific finite n, we still need the full formula with the (1 - $r^n$) term. Choice A correctly states this: even though |1/4| < 1 means the infinite series converges, calculating S_n for finite n still requires the standard formula. Choice B incorrectly claims S_n = a/(1 - r) for finite n (that's only the infinite sum); choice C wrongly states the series is infinite so S_n can't be computed (we can compute finite partial sums); choice D has the wrong exponent in the formula. Understanding this distinction is crucial: convergence of the infinite series doesn't change how we calculate finite partial sums!

This question tests your understanding of geometric series—the sum of terms from a geometric sequence—and how to apply the formula $S_n = a(1 - r^n) / (1 - r)$ to calculate these sums efficiently. A geometric series is the sum of terms from a geometric sequence: here the sequence is 3, 6, 12, 24, 48, 96 (multiply by 2 each time), the series is 3 + 6 + 12 + 24 + 48 + 96 (adding those terms). Rather than adding manually (tedious for many terms!), we use the formula $S_n = a(1 - r^n) / (1 - r)$, where a is first term, r is common ratio, and n is number of terms being summed. For this series, plug in a=3, r=2, n=6: $S_6 = 3(1 - 2^6)/(1 - 2) = 3(1 - 64)/(-1) = 3(-63)/(-1) = 189$; you can verify by adding step-by-step to build confidence. Choice A correctly applies the formula with accurate parameter identification and calculation, yielding 189. Choice C uses just the last term 96, perhaps forgetting to sum all terms—remember, the series is the sum, not a single term! The three-step geometric series strategy: (1) Identify a, r, n, (2) Check r ≠ 1 to avoid division by zero, (3) Substitute and calculate carefully, watching signs for r > 1. Understanding this formula turns tedious addition into quick computation—great job practicing!

This question tests your understanding of geometric series—the sum of terms from a geometric sequence—and how to derive and use the formula S sub n = a(1 - r to the n) divided by (1 - r) to calculate these sums efficiently. A geometric series with first term a = 7 and common ratio r = 1/2 gives us the sequence 7, 7/2, 7/4, 7/8, ... where each term is half the previous one. Rather than computing fractions and adding, we use the formula S sub n = a(1 - r to the n) divided by (1 - r). To find S sub 8 (sum of first 8 terms): substitute a = 7, r = 1/2, n = 8 into the formula. S sub 8 = 7(1 - (1/2) to the 8) divided by (1 - 1/2) = 7(1 - 1/256) divided by (1/2) = 7(255/256) divided by (1/2) = 7(255/256) × 2 = 1785/128. Choice A correctly shows this calculation and result. Choice B incorrectly uses n = 7 instead of n = 8—when finding S sub 8, we need 8 terms, so use (1/2) to the 8, not (1/2) to the 7. Always match the subscript on S to the exponent in the formula! The three-step geometric series strategy: (1) Identify parameters from the problem statement. (2) Substitute carefully into S sub n = a(1 - r to the n) divided by (1 - r). (3) Simplify fractions by finding common denominators. When r < 1, the series converges quickly—(1/2) to the 8 = 1/256 is tiny, so S sub 8 is close to the infinite sum 7/(1 - 1/2) = 14!

This question tests your understanding of geometric series—the sum of terms from a geometric sequence—and how to derive and use the formula $S_n = \dfrac{a(1 - r^n)}{1 - r}$ to calculate these sums efficiently. We can verify the formula by computing $S_4 = 3 + 6 + 12 + 24$ both manually ($3 + 6 + 12 + 24 = 45$) and using the formula. For the formula method: identify $a = 3$ (first term), $r = 2$ (each term doubles: $6/3 = 2$), and $n = 4$ (four terms). To apply the formula: $S_4 = 3(1 - 2^4)$ divided by $(1 - 2) = 3(1 - 16)$ divided by $(-1) = 3(-15)$ divided by $(-1) = 45$. This matches our manual calculation, verifying the formula works! Choice A correctly shows this calculation and result. Choice B incorrectly uses $n = 3$ instead of $n = 4$—remember $S_4$ means sum of 4 terms, so use $2^4 = 16$, not $2^3 = 8$. The subscript on S tells you the value of n! The three-step verification strategy: (1) Calculate manually as a check (here $3 + 6 + 12 + 24 = 45$). (2) Apply formula with correct parameters. (3) Verify both methods give the same answer. This builds confidence in the formula and catches errors!

This question tests your understanding of geometric series—the sum of terms from a geometric sequence—and adapting the formula S_n = a(1 - $r^n$) / (1 - r) for applications like bouncing ball distances. The total distance for 5 bounces is sum_${k=1}^5$ $210(0.8)^k$ = 20 sum_${k=1}^5$ $(0.8)^k$, where sum = 0.8(1 - $0.8^5$)/(1 - 0.8) since it starts from k=1, not 0. Rather than calculating each bounce, the formula handles the pattern: adjust to a=20*0.8=16, then $S=16(1-0.8^5$)/(1-0.8). Cancellation in derivation makes it reliable for decaying ratios like 0.8<1. Choice B correctly sets a=16, r=0.8, n=5 for the adjusted series. Choice A uses 20 as a but includes an extra term like sum from k=0, overestimating—remember to shift for starting at k=1! Factor out the first multiplier, confirm n by listing a few terms, and note |r|<1 means terms shrink—you're mastering these practical uses, great job!

This question tests your understanding of geometric series—the sum of terms from a geometric sequence—and how to apply the formula S_n = a(1 - $r^n$)/(1 - r) to calculate these sums efficiently. A geometric series is the sum of terms from a geometric sequence: here we have 3, 6, 12, 24, 48, 96 where each term is multiplied by 2 to get the next. Rather than adding manually (3 + 6 + 12 + 24 + 48 + 96), we use the formula with a = 3 (first term), r = 2 (common ratio), and n = 6 (number of terms). Substituting into S_6 = 3(1 - $2^6$)/(1 - 2) = 3(1 - 64)/(-1) = 3(-63)/(-1) = 189. Choice A correctly calculates this sum as 189. Choice B (192) might come from miscounting terms or arithmetic error, while C (93) appears to use only 5 terms instead of 6. The three-step geometric series strategy: (1) Identify a = 3, r = 2, n = 6, (2) Check r ≠ 1 (it's 2, so we're good), (3) Substitute carefully into the formula. Watch the signs: when r > 1, both (1 - $r^n$) and (1 - r) are negative, making the quotient positive!