Geometric Progression

The key distinction is between an arithmetic pattern with a constant difference and a geometric pattern with a constant ratio. In option B, each term is obtained by multiplying the preceding term by 2: 4 × 2 = 8, 8 × 2 = 16, and 16 × 2 = 32. Thus it follows a multiplication rule. Option A follows addition of 3 at every step, option C follows subtraction of 3, and option D follows addition of 4. These three sequences have constant differences rather than a repeated multiplier. Therefore option B is the only sequence formed by multiplication. The fact that its terms double each time, rather than increasing by a fixed amount, identifies it as a geometric progression.

The governing concept is a geometric progression, because each term is obtained by multiplying the preceding term by the same non-zero constant. The common ratio is r=12/4=3; this is confirmed by 36/12=3 and 108/36=3. For a geometric progression, the nth term is a_n=a_1r^(n−1). Therefore a_7=4×3^6=4×729=2916. Continuing the sequence also gives a_5=324, a_6=972, and a_7=2916. Hence option C is correct. Option A is the sixth term, while options B and D do not result from repeatedly multiplying by the fixed ratio 3. The initial term and ratio are both used, so no addition-based rule is appropriate.

The governing concept is a geometric progression, because the ratio of every term to the preceding term is constant. Here the common ratio is \(r=\frac13\). Starting from the first term, divide by 3 repeatedly: the first four terms are 81, 27, 9, and 3; the fifth term is \(3\div3=1\), and the sixth term is \(1\div3=\frac13\). Using the formula gives the same result: \(a_6=81(\frac13)^5=\frac{81}{243}=\frac13\). Therefore option B is correct. Option A is only the fifth term, option C applies the division one extra time, and option D is impossible because repeated division of a positive number by 3 never produces exactly zero.

The governing concept is a geometric progression, because every term is obtained by multiplying the previous term by the same ratio, 2. With first term 3 and common ratio 2, the nth term is a_n = 3×2^(n−1). To find the first term greater than 500, examine the boundary terms. The eighth term is a_8 = 3×2^7 = 3×128 = 384, which is not greater than 500. The ninth term is a_9 = 3×2^8 = 3×256 = 768, which is greater than 500. Since the progression is increasing, a_9 is the first term satisfying the condition. Therefore option B is correct. Checking only 768 would show that it is large enough, but comparing it with a_8 proves that it is the first such term.

The governing concept is the nth-term formula for a geometric progression. The first term is 5 and the common ratio is 2, so a_n = 5×2^(n−1). The exponent is n−1 because the first term has undergone zero doublings, the second has undergone one doubling, and so forth. To locate 5120, divide by the first term: 5120 ÷ 5 = 1024. Since 1024 = 2^10, we have 5120 = 5×2^10. Comparing this with 5×2^(n−1), n−1 = 10, so n = 11. Therefore option B is correct. Choosing the 10th term incorrectly uses n instead of n−1; choosing 12th or 13th counts extra doublings.

The governing concept for a geometric progression is a constant ratio between every pair of consecutive terms. To find that ratio, divide a term by the term immediately before it. Here, 6 divided by 2 equals 3; 18 divided by 6 also equals 3; and 54 divided by 18 again equals 3. Since the quotient remains constant, the common ratio is r = 3. Therefore option B is correct. Option A is the first term, not the ratio. Option C is obtained by confusing multiplication or subtraction with division, and option D does not describe the relationship between any consecutive terms. The sequence is geometric because each term is three times the preceding term.

Each term is multiplied by (2), so the next term is (40\times2=80). In exams, first identify the common ratio.

In a geometric progression, each term after the first is obtained by multiplying the preceding term by the common ratio. Therefore, to find the second term, multiply the first term by the given ratio. This rule is different from an arithmetic progression, where a fixed number is added instead.

Here the first term is \(4\) and the common ratio is \(2\). Hence the second term is \(4\times2=8\). The sequence begins \(4,8,16,32,\ldots\). Thus, option C is correct. The value 6 would come from adding 2, and 10 would come from adding 6; neither follows the geometric-progression rule given here.

The ratio of consecutive terms is (27\div81=\frac{1}{3}). In exams, a decreasing geometric progression can have a fractional ratio.

A sequence is a geometric progression when the ratio of each term to the preceding term is constant. Check the consecutive quotients: 6 divided by 3 equals 2, 12 divided by 6 equals 2, and 24 divided by 12 equals 2. Because every checked ratio is the same, the sequence is geometric and its common ratio is 2. Hence option A is correct. Option B confuses the ratio with the first term or with an unrelated multiplier. Option C is false because the ratios are equal, and option D is false because the terms are increasing, not decreasing. The repeated multiplication by 2 is the defining feature here.

The defining condition for a geometric progression is a constant multiplicative ratio between each pair of consecutive nonzero terms. In this sequence, 4 ÷ 2 = 2, 6 ÷ 4 = 3/2, and 8 ÷ 6 = 4/3. These ratios are different, so the sequence does not have one common ratio and is not geometric. Therefore option C is correct. The sequence is instead an arithmetic progression because its consecutive differences are constant: 4 - 2 = 2, 6 - 4 = 2, and 8 - 6 = 2. Having first term 2, increasing terms, or even terms does not decide whether a sequence is geometric; those are only observations. The decisive test is equality of consecutive ratios, not equality of differences.

The governing concept is identification of the first term in a geometric progression. When a sequence is written from left to right, its first term is the number at the beginning, usually denoted by a. The given progression begins with 7, so its first term is a = 7. The repeated multiplication by 3 is a separate property: 21 ÷ 7 = 3, 63 ÷ 21 = 3, and 189 ÷ 63 = 3, showing that the common ratio is 3. It does not alter the first term. Hence option B is correct. Options C and D are the second and third terms, respectively, while option A is the reciprocal of the common ratio, not a term of the displayed progression.

In a geometric progression, each term is obtained by multiplying the preceding term by the same common ratio r. If the first term is a, the terms begin as a, ar, and then \(ar^2\). Therefore the third term is found using the ratio twice, not once. With a=3 and r=4, the third term is \(ar^2=3\times4^2=3\times16=48\).

The sequence can also be written step by step: the first term is 3, the second is \(3\times4=12\), and the third is \(12\times4=48\). Hence option C is correct. The value 12 is the second term, while 24 does not follow the required multiplication pattern. The value 64 is only \(4^3\), so it ignores the first term 3. The exponent in the formula is one less than the term number.

Each term is multiplied by (\frac{1}{2}), so the next term is (8\times\frac{1}{2}=4). In exams, identify fractional ratios in decreasing sequences.

The common ratio is (3), so the fifth term is (54\times3=162). In exams, multiply the last known term by the ratio.

The ratio of (36) and (108) is (3), so the missing term is (36\div3=12). In exams, apply the ratio backward too.

Here, the first term is \(a=3\) and the common ratio is \(r=3\). Thus, \(a_n=ar^{n-1}=3\times3^{n-1}=3^n\). The option \(a_n=3^{n-1}\) gives 1 as the first term, not 3. Exam tip: substitute \(n=1\) to check whether a proposed general term gives the first term of the sequence.

Given \(a_n=2^n\). For the fifth term, substitute \(n=5\): \(a_5=2^5=32\). Therefore, 32 is correct. The value 16 equals \(2^4\), so it represents the fourth term, not the fifth. Exam tip: To find a particular term of a sequence, substitute its term number directly for \(n\) in the given formula.

The first term is (5) and the ratio is (3), so (a_n=5\cdot3^{n-1}). In exams, use (a_n=ar^{n-1}).

The governing concept is the term-to-term rule of a geometric progression: each term is obtained by multiplying the preceding term by the same common ratio. Here a_1 = 18 and r = 1/3. Therefore a_2 = a_1r = 18 × 1/3 = 6, so option B is correct. This can also be checked from the general formula a_n = a_1r^(n-1): for n = 2, a_2 = 18(1/3)^1 = 6. Because the ratio is less than 1, the second term should be smaller than 18, which rules out 9 and 12 as well as any larger value. Option 3 would require an incorrect division by 6 rather than multiplication by one-third. Thus the ratio operation and the direct formula agree.