Geometric Progression

Correct answer: A. (\frac{601}{3})

Explanation: (a_1=\frac{22}{3}) and (a_4=198), so the sum is (\frac{616}{3}). The correct option should include (\frac{616}{3}).

Correct answer: C. (60)

Explanation: (x^2=36\cdot100=3600), so positive (x=60). The square of the middle term equals the product of the outer terms.

Correct answer: C. (2500)

Explanation: The direct answer is option C: \(a_5=2500\). For a geometric progression, \(a_n=a_1r^{n-1}\). The fourth term is three ratio steps after the first, so \(a_4=4r^3\). Using \(a_4=500\), we get \(500=4r^3\), hence \(r^3=125\). Because the ratio is positive, \(r=5\). The next term is therefore \(a_5=a_4r=500\times5=2500\). Option A, 1500, does not use the ratio obtained from the given terms. Option B, 2000, is also not the result of multiplying 500 by 5. Option C is correct because it follows directly from the verified ratio. Option D, 3000, would require a ratio of 6, which would make the fourth term inconsistent with the data. A quick check is the sequence \(4,20,100,500,2500\), where every term is five times the previous one. Memory cue: when the first and fourth terms are known, solve for \(r^3\), then multiply the fourth term by \(r\).

Correct answer: C. 5120

Explanation: The nth term of a geometric progression is \(a_n=ar^{n-1}\). Therefore, \(a_6=5\times4^{6-1}=5\times4^5=5\times1024=5120\). Hence, option C is correct. \(2560\) results from an error in evaluating the power of 4 or in multiplication. Exam tip: the exponent in the nth term is always \(n-1\), not \(n\).

Correct answer: B. (4)

Explanation: Here \(r=\frac{1}{2}\), so \(a_9=1024\cdot\left(\frac{1}{2}\right)^8=4\). In exams, apply fractional ratios carefully in decreasing GPs.

Correct answer: B. (9)th term

Explanation: From (7\cdot2^{n-1}=1792), (2^{n-1}=256=2^8), so (n=9). In exams, equate powers to find the term number.

Correct answer: B. 3

Explanation: The nth term of a geometric progression is aₙ = arⁿ⁻¹. Hence, for the fifth term, 4r⁴ = 324. So, r⁴ = 81 = 3⁴. Since r is stated to be positive, r = 3. Although r = −3 also gives r⁴ = 81, it is not positive. Exam tip: in the fifth term, the power of r is 4, not 5.

Correct answer: A. (a_n=6\cdot4^{n-1})

Explanation: The first term is (6) and the ratio is (4), so (a_n=6\cdot4^{n-1}). In exams, keep (a) and (r) correct in (ar^{n-1}).

Correct answer: C. (36)

Explanation: For the middle term, (x^2=12\cdot108=1296), so (x=36). In exams, the square of the middle GP term equals the product of the outer terms.

Correct answer: C. 2040

Explanation: Here, the first term is \(a=8\), the common ratio is \(r=2\), and the number of terms is \(n=8\). The sum of the first \(n\) terms of a geometric progression is \(S_n=\frac{a(r^n-1)}{r-1}\). Thus, \(S_8=\frac{8(2^8-1)}{2-1}=8(256-1)=2040\). Therefore, option C is correct. Option 4080 incorrectly doubles the sum. Exam tip: identify \(a\), \(r\), and \(n\) before applying the GP sum formula.

Correct answer: A. \(a_n^2=a_{n-1}a_{n+1}\)

Explanation: In a GP, \(a_{n-1}=a_n/r\) and \(a_{n+1}=a_nr\). Multiplying gives \(a_{n-1}a_{n+1}=a_n^2\). Options B and D describe the equal-difference property of an AP. Exam tip: square the middle term to test a GP quickly.

Correct answer: B. \(a_n=243\cdot\left(\frac{1}{3}\right)^{n-1}\)

Explanation: The first term is (243) and the ratio is \(\frac{1}{3}\), so the correct rule is \(243\cdot\left(\frac{1}{3}\right)^{n-1}\). In exams, write the fractional ratio in a decreasing GP.

Correct answer: A. \(2\)

Explanation: The \(n\)th term of a GP is \(a_n=ar^{n-1}\). Thus, \(a_6=3r^5=96\), so \(r^5=32=2^5\). Since \(r\) is positive, \(r=2\). If \(r=3\), then \(3\times3^5\) is not equal to 96. Exam tip: for the sixth term, the exponent of \(r\) is \(6-1=5\).

Correct answer: B. (6)th term

Explanation: The direct answer is option B: 5120 is the sixth term. In a geometric progression, each term is obtained by multiplying the previous term by the same number, called the common ratio. Here, the terms are 5, 20, 80, 320, so the ratio is 4. The nth-term formula is \(a_n=a_1r^{n-1}\). Therefore \(5120=5\times4^{n-1}\). Dividing by 5 gives \(1024=4^{n-1}\). Since \(1024=4^5\), we get \(n-1=5\), so \(n=6\). A direct check also helps: first 5, second 20, third 80, fourth 320, fifth 1280, sixth 5120. Option A is wrong because the fifth term is 1280. Option B is correct because the sixth term is 5120. Option C is wrong because the seventh term would be \(5120\times4=20480\). Option D is wrong because the eighth term would be \(81920\). The important idea is that the exponent in a GP nth-term formula is one less than the term number. Memory cue: count the first term as exponent zero, not exponent one.

Correct answer: A. (5)

Explanation: \(a_6=160\cdot\left(\frac{1}{2}\right)^5=5\). In exams, apply powers of fractional ratios carefully.

Correct answer: A. (2)

Explanation: (\frac{a_8}{a_4}=r^4=\frac{768}{48}=16), so (r=2). In exams, the ratio of distant terms gives a power based on the position gap.

Correct answer: B. \(5, 10, 20, 35\)

Explanation: In a GP, the ratio of every term to its preceding term must remain constant. In option B, \(10/5=2\) and \(20/10=2\), but \(35/20=7/4\). Hence it is not a GP. Exam tip: check all consecutive ratios.

Correct answer: B. (81)

Explanation: The position gap is (4) and (r=3), so the ratio is (3^4=81). In exams, use (\frac{a_m}{a_n}=r^{m-n}).

Correct answer: A. (2295)

Explanation: Here (a=9) and (r=2), so (S_8=9(2^8-1)=2295). In exams, identify (a) and (r) from the general term.

Correct answer: B. (6)th term

Explanation: From (3\cdot5^{n-1}=9375), (5^{n-1}=3125=5^5), so (n=6). In exams, compare powers to find the term number.

Correct answer: B. (3)

Explanation: (\frac{a_6}{a_2}=r^4=\frac{1458}{18}=81), so (r=3). In exams, a gap of (4) positions gives (r^4).

Correct answer: A. \(2\)

Explanation: In a geometric progression, the seventh term is \(a_7=ar^6\). Thus, \(384=6r^6\), so \(r^6=64=2^6\). Since \(r\) is stated to be positive, \(r=2\). Although \(r=-2\) also satisfies \(r^6=64\), it violates the given condition. Exam tip: in \(a_n=ar^{n-1}\), the exponent of \(r\) is always \(n-1\).

Correct answer: A. 6558

Explanation: Here, the first term is \(a=6\), the common ratio is \(r=3\), and the number of terms is \(n=7\). The sum of the first \(n\) terms of a GP is \(S_n=\frac{a(r^n-1)}{r-1}\). Therefore, \(S_7=\frac{6(3^7-1)}{3-1}=\frac{6(2187-1)}{2}=6558\). Hence, option A is correct. A nearby value such as \(6560\) can result from an arithmetic error in the final calculation. Exam tip: write down \(a\), \(r\), and \(n\) separately before applying the formula.

Correct answer: B. (2)

Explanation: (\frac{6250}{50}=125=r^3), so (r=5) and (a_1=\frac{50}{5^2}=2). In exams, find (r) first and then the first term.

Correct answer: D. (5)

Explanation: The fifth term is (x\cdot5^4=625x=3125), so (x=5). In exams, put the algebraic first term in (ar^{n-1}) too.