Geometric Progression

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.

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.

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.

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.

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.

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).

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.

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.

(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.

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.

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

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.

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

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.

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 \(b^2=ac\).

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 \(n\).

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

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 \(n-1\).

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.

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.

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.

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.

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.

(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.

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.