Geometric Progression

Correct answer: B. 18

Explanation: The governing property is that consecutive terms of a geometric progression have one constant ratio. From the first two terms, r = 6 ÷ 2 = 3. The missing third term must therefore be x = 6 × 3 = 18. We can verify the answer using the final term: 18 × 3 = 54, so the complete sequence is 2, 6, 18, 54. Hence option B is correct. Option A would give a ratio of 2 between the second and third terms and would produce 36 rather than 54 next. Option C would give a ratio of 4, and option D would give a ratio of 5; neither agrees with the established ratio 3. Both forward construction and backward checking lead to 18.

Correct answer: C. \((5,-10,20,-40,\ldots)\)

Explanation: A negative common ratio means that each term is obtained by multiplying the previous term by a negative number, so the signs alternate when the terms are nonzero. In option C, \(\frac{-10}{5}=-2\), \(\frac{20}{-10}=-2\), and \(\frac{-40}{20}=-2\). Thus its common ratio is negative, and option C is correct. Option A has ratio 2, option B has ratio 3, and option D has ratio \(\frac12\); all three ratios are positive. A decreasing sequence does not automatically have a negative ratio, as option D demonstrates.

Correct answer: B. 150

Explanation: The first four terms are 10, 20, 40, and 80. Therefore, their sum is 10+20+40+80=150. This is a GP with first term 10 and common ratio 2; using the formula also gives \(S_4=10(2^4-1)/(2-1)=150\). A result such as 160 comes from an incorrect addition. Exam tip: for a few terms, verify the answer by direct addition.

Correct answer: A. The ratio of every pair of consecutive terms is constant

Explanation: In a GP, each term is obtained by multiplying the previous term by the same constant, so consecutive-term ratios remain equal. For 3, 6, 12, the ratio is 2. Equal differences describe an AP. Exam tip: check ratios first.

Correct answer: A. (a_n=6\cdot3^{n-1})

Explanation: The first term is (6) and the ratio is (3), so (a_n=6\cdot3^{n-1}). Use both first term and ratio in the general term.

Correct answer: B. 63

Explanation: This is a geometric progression with first term a=1, common ratio r=2, and n=6. Thus, S_n=a(r^n-1)/(r-1)=1(2^6-1)/(2-1)=63. The terms are 1, 2, 4, 8, 16, and 32, whose sum is 63. Note that 64 is only 2^6, not the sum. Exam tip: for r≠1, use S_n=a(r^n-1)/(r-1).

Correct answer: C. (100)

Explanation: The common ratio is (\frac{20}{4}=5), so (x=20\cdot5=100). Multiply the second term by the ratio to get the third term.

Correct answer: B. Fifth term

Explanation: This is a geometric progression with first term 2 and common ratio 4. Its terms are 2, 8, 32, 128, 512. Therefore, 512 is the fifth term. The fourth term is 128, so it is not correct. Exam tip: In a GP, multiply each term by the common ratio to obtain the next term.

Correct answer: B. (3)

Explanation: (a_4=a_2r^2), so (162=18r^2) and (r^2=9), giving positive ratio (r=3). Use the gap between terms as the exponent.

Correct answer: B. 16

Explanation: The governing concept is the constant-ratio rule of a geometric progression. Dividing consecutive terms gives r = −2 ÷ 1 = −2, 4 ÷ (−2) = −2, and −8 ÷ 4 = −2. Thus the next term is obtained by multiplying the fourth term by −2: a₅ = (−8)(−2) = 16. Therefore, option B is correct. The negative ratio makes the signs alternate: positive, negative, positive, negative, and then positive. Option A has the wrong magnitude and would not result from multiplying −8 by −2. Option C keeps the sign negative, ignoring the product of two negatives, while option D uses an incorrect magnitude. The constant ratio gives a direct and reliable check.

Correct answer: C. (9,27,81,243)

Explanation: The first term is (9), and multiplying by (3) each time gives (9,27,81,243). Match both (a) and (r).

Correct answer: A. 3375

Explanation: The first three terms are 5, 15, and 45. Their product is \(5\times15\times45=75\times45=3375\). Note that 2025 equals \(45\times45\), so it is not the product of the first three terms. Exam tip: First list the required terms, then multiply them in order.

Correct answer: B. (\frac{25}{2})

Explanation: The common ratio is (\frac{1}{2}), so the next term is (25\cdot\frac{1}{2}=\frac{25}{2}). Multiply by the ratio.

Correct answer: C. (4,8,16,32,\ldots)

Explanation: The rule \\(a_n=4\\cdot2^{n-1}\\) gives each term of the sequence. Start with \\(n=1\\): \\(a_1=4\\cdot2^0=4\\). For \\(n=2\\), \\(a_2=4\\cdot2^1=8\\); for \\(n=3\\), \\(a_3=4\\cdot2^2=16\\); and for \\(n=4\\), \\(a_4=4\\cdot2^3=32\\). Thus the terms begin 4, 8, 16, 32, and each term is twice the preceding term.

This exactly matches option C. Option A increases by addition, so it is an arithmetic pattern rather than this geometric pattern. Options B and D have the same doubling idea but start with 2 and 8, respectively, not 4. Therefore, option C follows directly from substituting consecutive positive integer values of n.

Correct answer: C. 4

Explanation: In a geometric progression, the ratio of consecutive terms remains the same. Here, the common ratio is \(r=\frac{36}{12}=3\). Hence, \(\frac{12}{x}=3\), so \(x=\frac{12}{3}=4\). If 6 were chosen, the ratios would be \(\frac{12}{6}=2\) and \(\frac{36}{12}=3\), which are not equal. Exam tip: for three consecutive terms \(a,b,c\) of a GP, you may also use \(b^2=ac\).

Correct answer: C. 1944

Explanation: The first term is \(a=8\) and the common ratio is \(r=24/8=3\). The \(n\)th term of a GP is \(a_n=a r^{n-1}\). Therefore, \(a_6=8\times3^{6-1}=8\times243=1944\). Note that 5832 is the next, or seventh, term because \(1944\times3=5832\). Exam tip: for the \(n\)th term, use the exponent \(n-1\), not \(n\).

Correct answer: B. (2,10,50,250,\ldots)

Explanation: In (2,10,50,250,\ldots), each term is multiplied by (5). For the ratio divide the next term by the previous term.

Correct answer: C. \(6\)

Explanation: In a geometric progression, the \(n\)th term is \(a_n=a_1r^{n-1}\). Therefore, \(a_4=\frac{3}{4}\times2^{4-1}=\frac{3}{4}\times8=6\). Option \(8\) is only the value of \(2^3\); it must also be multiplied by the first term \(\frac{3}{4}\). Exam tip: the exponent in \(a_n\) is always \(n-1\).

Correct answer: C. 62

Explanation: The first term of this GP is 2 and the common ratio is 2. Its first five terms are 2, 4, 8, 16, and 32, so their sum is \(2+4+8+16+32=62\). Therefore, 62 is correct. The value 32 is only the fifth term, not the sum. Exam tip: Count the terms carefully; 32 comes after 16 as the fifth term.

Correct answer: B. 12

Explanation: For three consecutive terms of a geometric progression, the square of the middle term equals the product of the first and third terms. Thus, \(x^2=8\times18=144\). Since the progression is positive, \(x=\sqrt{144}=12\); \(-12\) is not allowed. Values such as 10 or 14 do not give \(x^2=144\). Exam tip: the middle of three consecutive GP terms is the positive geometric mean of the outer terms.

Correct answer: A. (a_n=7\cdot3^{n-1})

Explanation: The first term is (7) and the ratio is (3), so (a_n=7\cdot3^{n-1}). Use the first term and ratio in the general term.

Correct answer: A. (2)

Explanation: For a geometric progression, moving from the second term to the fifth term requires three multiplications by the common ratio. Thus \(a_5=a_2r^3\). Substituting the given values gives \(192=24r^3\). Dividing both sides by 24 gives \(r^3=8\). Since the problem states that the ratio is positive, the real value is \(r=2\), because \(2^3=8\). Therefore option A is correct.

The position gap explains the exponent: from term 2 to term 3 is one step, to term 4 is two steps, and to term 5 is three steps. A ratio of 3 would give \(3^3=27\), not 8, while 4 and 8 are also inconsistent with the equation. The positivity condition removes any concern about a negative cube root; in fact, the real cube root of 8 is uniquely 2.

Correct answer: C. 768

Explanation: Direct answer: a₅ = 768, so option C is correct. In a geometric progression, each term is obtained by multiplying the preceding term by the same common ratio. Here r = 12 ÷ 3 = 4, and 48 ÷ 12 = 4 confirms the ratio. The general term is aₙ = a₁rⁿ⁻¹. For the fifth term, the ratio is used four times after the first term: a₅ = 3 × 4⁴ = 3 × 256 = 768. We can also build the terms: a₁ = 3, a₂ = 12, a₃ = 48, a₄ = 192, and a₅ = 768. Option A is only the fourth term. Option B is not produced by multiplying 192 by 4 and results from an arithmetic or power error. Option C follows both the formula and the repeated-multiplication rule. Option D would require a different ratio and does not fit the progression. Memory cue: for aₙ, the ratio is used n − 1 times, not n times.

Correct answer: C. (5)

Explanation: (a_3=a_1r^2), so (50=2r^2) and (r^2=25), hence positive (r=5). When positive ratio is asked take the positive root.

Correct answer: B. It is a geometric progression with (r=\frac{1}{2})

Explanation: Each term is half of the previous term, so (r=\frac{1}{2}). To check the ratio divide the next term by the previous term.