Geometric Progression

The common ratio of a geometric progression is the factor used to multiply one term to get the next term. It can be found by dividing a term by the preceding term. Using the first two terms here gives \(r=\frac{12}{4}=3\). The same factor is confirmed by \(\frac{36}{12}=3\) and \(\frac{108}{36}=3\), so the sequence is consistently geometric.

Thus the correct answer is option B. Multiplying 4 by 3 gives 12, multiplying 12 by 3 gives 36, and multiplying 36 by 3 gives 108. The values 2, 4, and 6 do not reproduce these consecutive terms: for example, a ratio of 2 would make the second term 8, not 12. The ratio is a multiplier, not the difference between terms.

Each term is multiplied by (2) so the next term is (56\times2=112). In exams first identify the common ratio.

The governing property of a geometric progression is multiplicative: every term after the first is obtained by multiplying the preceding term by the common ratio. Here a₁ = 5 and r = 3, so a₂ = a₁r = 5 × 3 = 15. Therefore, option D is correct. Option A comes from adding 3 to 5, which is the type of operation associated with an arithmetic progression, not a geometric one. Option B would result from multiplying by 2, and option C does not use the stated ratio in any valid way. The defining operation is multiplication by 3, so the sequence would begin 5, 15, 45, 135, and so on. This confirms that 15 is the only suitable answer.

The ratio of consecutive terms is (25\div125=\frac{1}{5}). In exams a decreasing geometric progression can have a fractional ratio.

The governing test for a geometric progression is that the ratio of every pair of consecutive terms must be the same. Compute the ratios: 8 ÷ 2 = 4, 32 ÷ 8 = 4, and 128 ÷ 32 = 4. Since the common ratio is consistently 4, the sequence is a geometric progression. Therefore, option B is correct. Option A identifies the wrong ratio; the terms are multiplied by 4, not 2. Option C is false because all the calculated ratios agree. Option D is also false because the terms increase from 2 to 128. A sequence may be increasing or decreasing and still be geometric; equality of consecutive ratios is the decisive condition.

A geometric progression is defined by a constant ratio between every pair of consecutive terms. For this sequence, the first ratio is 6 ÷ 3 = 2, the second is 9 ÷ 6 = 3/2, and the third is 12 ÷ 9 = 4/3. Since these ratios are unequal, the sequence cannot be geometric, so option C is correct. In fact, it is an arithmetic progression because the consecutive differences are constant: 6 − 3 = 3, 9 − 6 = 3, and 12 − 9 = 3. The fact that the terms increase does not rule out a geometric progression; for example, 2, 4, 8 increases geometrically. Likewise, having first term 3 or having multiples of 3 is irrelevant to the defining ratio.

In a geometric progression, the first term is simply the first number written in the ordered sequence. The common ratio and later terms are useful for describing the pattern, but they do not change the identity of the first term. In the sequence (9,27,81,243,dots), the sequence begins with 9.

Thus a=9, and option B is correct. The pattern also confirms this: each term is obtained by multiplying the previous term by 3, since 27/9=3 and 81/27=3. However, this calculation is not needed to identify the first term. The displayed alternatives use fractions such as 9/9, but the intended listed value is the initial term 9, not a ratio or a quotient.

Each term is multiplied by (\frac{1}{3}) so the next term is (6\times\frac{1}{3}=2). In exams identify fractional ratios in decreasing sequences.

The common ratio is (4) so the fifth term is (192\times4=768). In exams multiply the last known term by the ratio.

The ratio of (45) and (135) is (3) so the missing term is (45\div3=15). In exams apply the ratio backward too.

The direct answer is A: \\(a_n=2\cdot4^{n-1}\\). For a geometric progression, the general term is \\(a_n=ar^{n-1}\\), where \\(a\\) is the first term and \\(r\\) is the common ratio. Here the first term is 2, and the ratios are \\(8/2=4\\), \\(32/8=4\\), and \\(128/32=4\\). Thus \\(a=2\\) and \\(r=4\\), giving \\(a_n=2\cdot4^{n-1}\\). A is correct. B, \\(4n-2\\), is a linear expression; it gives 2 for \\(n=1\\) but gives 6 for \\(n=2\\), not 8. C, \\(2^n\\), gives 2, 4, 8, not the given sequence. D, \\(8n\\), gives 8 for the first term, not 2. The exponent starts at \\(n-1\\) so that the first term has factor \\(4^0=1\\). Exam cue: identify first term and common ratio before applying the GP formula.

Given \(a_n=5^n\), substitute \(n=3\): \(a_3=5^3=5\times5\times5=125\). The value \(25\) is \(5^2\), so it results from using the wrong index. Exam tip: substitute the stated term number directly for \(n\) before evaluating the power.

The first term is (6) and the ratio is (3) so (a_n=6\cdot3^{n-1}). In exams use (a_n=ar^{n-1}).

The governing rule is a₂ = a₁r: to obtain the next term of a geometric progression, multiply the current term by the common ratio. Here a₁ = 40 and r = 1/2, so a₂ = 40 × 1/2 = 20. Therefore, option B is correct. Because the ratio is a fraction less than 1, the sequence decreases, which is consistent with 40 becoming 20. Option A would result from halving 20 again, option C does not use the ratio correctly, and option D doubles the first term instead of multiplying by one-half. Careful treatment of the fraction is essential.

The ratios (3\div1=3) and (6\div3=2) are not equal. In exams equal ratio is necessary for a geometric progression.

The governing concept is the common ratio of a geometric progression. The common ratio is obtained by dividing any term by the immediately preceding term, and the quotient must remain constant. Using the first two terms, r = 40 ÷ 10 = 4. This is confirmed by the next pairs: 160 ÷ 40 = 4 and 640 ÷ 160 = 4. Therefore, option B is correct. Option A would make the second term 20 rather than 40; option C would make it 80; and option D would make it 100. The first term being 10 does not mean that r is 10. The defining test is the repeated multiplier between consecutive terms, and here that multiplier is consistently 4 throughout the displayed progression.

The nth term of a geometric progression is \(T_n=ar^{n-1}\). Therefore, \(T_4=5\times3^{4-1}=5\times27=135\). The value 45 is the third term because \(5\times3^2=45\). Exam tip: for the fourth term, use the exponent \(4-1=3\).

The common ratio is (\frac{1}{2}) so the fifth term is (12\times\frac{1}{2}=6). In exams simplify fractions.

The terms are (2,6,18,54,162) so (162) is the fifth term. In exams list small terms in order to check.

For \(n=1\), \(a_1=4\cdot3^0=4\); for \(n=2\), \(a_2=4\cdot3^1=12\); and for \(n=3\), \(a_3=4\cdot3^2=36\). Therefore, the first three terms are \((4, 12, 36)\). In \((3, 9, 27)\), the first term is 3, but the given formula gives \(a_1=4\). Exam tip: substitute \(n=1\) first to verify the initial term.

The first term is (9) and the ratio is (4) so (a_n=9\cdot4^{n-1}). In exams match both the first term and the ratio.

The governing property is a constant common ratio between consecutive terms. From the first two terms, r = 15 ÷ 3 = 5; this is confirmed because 75 ÷ 15 = 5. To obtain the missing fourth term, multiply the third term by the same ratio: 75 × 5 = 375. Thus option C is correct. Option A would multiply 75 by 3, not by the established ratio; option B does not preserve the ratio; and option D would require a ratio of 6. Continuing a geometric progression means applying the same multiplier at every step, not adding a fixed number or choosing a convenient larger value.

The ratio of consecutive terms is (24\div48=\frac{1}{2}). In exams write the fractional ratio for a decreasing geometric progression.

After the fourth term (320) the fifth is (1280) and the sixth is (5120). In exams include the first term while counting.

The defining feature of a geometric progression is a constant multiplier between consecutive terms. To test the sequence, divide each term by the preceding term. If the same quotient is obtained each time, the sequence is geometric, and that quotient is its common ratio.

For this sequence, \(36\div12=3\), \(108\div36=3\), and \(324\div108=3\). Thus every term is obtained by multiplying the previous term by 3. The sequence is therefore a geometric progression with common ratio 3, so option A is correct. The number 12 is the first term, while the sequence is increasing, not decreasing.