Geometric Progression

Correct answer: B. (3, 18, 108, 648)

Explanation: In a geometric progression, each term is obtained by multiplying the preceding term by the common ratio. Here, the first term is 3 and r = 6: 3, 3×6 = 18, 18×6 = 108, and 108×6 = 648. Therefore, (3, 18, 108, 648) is correct. In option A, the terms are multiplied by 2, not by 6. Exam tip: Write the first term first, then multiply each successive term by r.

Correct answer: B. \(a_n=64\cdot\left(\frac{1}{2}\right)^{n-1}\)

Explanation: The first term is (64) and the ratio is \(\frac{1}{2}\) so \(a_n=64\cdot\left(\frac{1}{2}\right)^{n-1}\). In exams use the fractional ratio in a decreasing sequence.

Correct answer: A. \(a=2,\ r=5\)

Explanation: In a geometric progression, \(a\) is the first term, so \(a=2\). The common ratio \(r\) is the quotient of consecutive terms: \(r=\frac{10}{2}=5\). This is confirmed by \(50\div10=5\) and \(250\div50=5\). In option D, the ratio is correct, but the first term is 2, not 10. Exam tip: identify \(a\) from the first term and find \(r\) by dividing the second term by the first.

Correct answer: B. 3, 6, 12, 24, ...

Explanation: In a geometric progression, each term is obtained by multiplying the previous term by the same constant ratio. In option B, 6/3 = 12/6 = 24/12 = 2. Options A and D are arithmetic progressions because their common difference is constant. Exam tip: divide consecutive terms to check the common ratio.

Correct answer: B. Fifth term

Explanation: The terms in order are 3, 6, 12, 24, and 48. Therefore, 48 is the fifth term. The sixth term would be 96, so it is not correct. Exam tip: always count the given first term as term 1.

Correct answer: C. 4

Explanation: In a geometric progression, the common ratio is the ratio of consecutive terms. Thus, \(r=\frac{a_2}{a_1}=\frac{32}{8}=4\). Therefore, 4 is correct. The number 8 is the first term, not the common ratio. Exam tip: divide the second term by the first term to find the common ratio.

Correct answer: C. (176)

Explanation: The common ratio is (2) so the next term is (88\times2=176). In exams multiply the last known term by the ratio.

Correct answer: B. (10)

Explanation: The common ratio is (\frac{1}{2}) so the fifth term is (20\times\frac{1}{2}=10). In exams apply fractional ratios carefully.

Correct answer: C. 4

Explanation: Direct answer: the common ratio is 4, so option C is correct. In a geometric progression, divide any nonzero term by the term immediately before it. Using the first two terms, r = 4y/y = 4, provided y is not zero so that the ratio is defined. The next pair confirms this: 16y/(4y) = 4. Therefore every term is obtained by multiplying the previous term by 4: y × 4 = 4y and 4y × 4 = 16y. Option A would mean the terms remain unchanged, which they do not. Option B would give the second term as 2y, not 4y. Option C gives both displayed transitions correctly. Option D is the third coefficient, not the ratio between consecutive terms. A common confusion is to choose a number appearing in a term rather than calculate a quotient; always compare consecutive terms by division.

Correct answer: B. (1)

Explanation: Consecutive terms are equal so the ratio is (8\div8=1). In exams a non-zero constant sequence can also be a geometric progression.

Correct answer: C. (13,13,13)

Explanation: When the common ratio is (1) all terms remain (13). In exams (r=1) means a constant geometric progression.

Correct answer: B. (10)

Explanation: The ratio of (50) and (250) is (5) so the missing term is (50\div5=10). In exams apply the common ratio backward too.

Correct answer: A. (2)

Explanation: In the formula (a_n=7\cdot2^{n-1}) the common ratio is (2). In exams identify (r) in (ar^{n-1}).

Correct answer: C. 1134

Explanation: The common ratio of this GP is \(r=42/14=3\). Here, \(378\) is the fourth term, so \(a_5=378\times 3=1134\). \(945\) is not correct because it is not obtained by multiplying the previous term by 3. Exam tip: count the given terms first, then multiply the last given term by the common ratio.

Correct answer: C. 80

Explanation: The \(n\)th term of a geometric progression is \(a_n=a_1r^{n-1}\). Hence, \(a_3=5\times4^{3-1}=5\times16=80\). Therefore, 80 is correct. The close distractor 20 is the second term, \(a_2=5\times4\), not the third term. Exam tip: when finding the \(n\)th term, use \(n-1\) as the exponent of \(r\).

Correct answer: B. (81,27,9,3)

Explanation: In (81,27,9,3) each term is multiplied by (\frac{1}{3}). In exams write the ratio of a decreasing geometric progression as a fraction.

Correct answer: B. (2048)

Explanation: This is of the form (4\cdot2^{n-1}) so the tenth term is (4\cdot2^9=2048). In exams identify powers to solve quickly.

Correct answer: C. 64

Explanation: Given \(a_n=4^n\). Substituting \(n=3\), we get \(a_3=4^3=4\times4\times4=64\). The value \(16\) is \(4^2\), so it does not represent \(a_3\). Exam tip: when finding a term, substitute the given index correctly in the exponent.

Correct answer: A. \(a_n=36\cdot\left(\frac{1}{2}\right)^{n-1}\)

Explanation: The direct answer is A: \(a_n=36\cdot(1/2)^{n-1}\). In a geometric progression, each term is obtained by multiplying the previous term by one fixed ratio. The ratios are 18/36=1/2, 9/18=1/2 and (9/2)/9=1/2. Thus a=36 and r=1/2. Using \(a_n=ar^{n-1}\) gives the expression in option A. Substitution verifies it: the first four values are 36, 18, 9 and 9/2. Option B uses ratio 2, so it would increase to 72, 144 and so on. Option C, 36-n, has a constant difference of -1 and is not the required geometric pattern. Option D, 18n, gives 18, 36, 54, not the sequence. Therefore A is the only formula that works. Exam cue: a decreasing GP with each term half the previous one has ratio \(1/2\), not 2.

Correct answer: C. (24)

Explanation: The square of the middle term is (8\times72=576) so the positive (x=24). In exams for three GP terms the square of the middle term equals the product of the outer terms.

Correct answer: C. (3072)

Explanation: The common ratio is (4) so the fifth term is (768\times4=3072). In exams multiply the last known term by the ratio.

Correct answer: A. (6, 24, 96, 384, …)

Explanation: For a geometric progression, two conditions must hold: the first term must equal the stated initial value, and every following term must be obtained by multiplying by the stated common ratio. Starting with 6 and repeatedly multiplying by 4 gives 6 × 4 = 24, 24 × 4 = 96, and 96 × 4 = 384. Therefore, option A satisfies both requirements exactly. Option B starts with 4 and has ratio 2. Option C has a constant difference of 4, so it is an arithmetic progression rather than a geometric one. Option D has ratio 4, but it starts with 24 instead of 6; it begins at the second term of the required progression. Hence only option A is correct.

Correct answer: A. 3, 6, 12, 24, ...

Explanation: In a geometric progression, the ratio of each term to the preceding term remains constant. In option A, 6/3 = 12/6 = 24/12 = 2, so it is a GP. Option B has a constant difference, so it is an AP. Exam tip: check ratios of consecutive terms.

Correct answer: C. (3)

Explanation: In a geometric progression, the common ratio is the number by which each term is multiplied to obtain the next term. Compute the quotient of consecutive terms: 18/6=3. Checking further gives 54/18=3 and 162/54=3, so the same multiplier is used throughout the sequence. Therefore the common ratio is r=3.

Option C is consequently correct. The ratio is not found by subtracting terms; the differences are 12, 36, and 108, which are not constant. Nor should the first term itself be chosen as the ratio. Dividing any nonzero term by the preceding term gives the same result here, confirming that the sequence is geometric with positive ratio 3.

Correct answer: C. (1024)

Explanation: Each term is multiplied by (4) so the next term is (256\times4=1024). In exams first identify the common ratio.