Geometric Progression

The defining property of a geometric progression is that each term after the first is obtained by multiplying the preceding term by the same common ratio. Here the first term is a₁ = 7 and the common ratio is r = 4. Hence the second term is a₂ = a₁r = 7 × 4 = 28, so option C is correct. Adding 4 to 7 gives 11, but addition describes an arithmetic progression rather than a geometric one. The value 21 would require a ratio of 3, not 4, and 32 is not produced by applying the stated rule to the first term. Checking the ratio 28/7 = 4 confirms that 28 satisfies both conditions exactly.

The ratio of consecutive terms is (81\div243=\frac{1}{3}). In exams a decreasing geometric progression can have a fractional ratio.

A sequence is a geometric progression when the ratio of every term to the immediately preceding term is constant. Check the consecutive ratios here: 25/5 = 5, 125/25 = 5, and 625/125 = 5. Since all these ratios are equal, the sequence is a geometric progression with common ratio 5. Therefore, option A is correct. The number 25 is a term of the sequence, not the common ratio, so option B confuses a term with a multiplier. Option C is contradicted by the equal ratios, and option D is false because the terms increase from 5 to 625 rather than decrease. The defining test is equal consecutive ratios.

In a geometric progression, the ratio of each term to the preceding term must remain constant. Here, \(6\div2=3\), but \(12\div6=2\) and \(20\div12=\frac{5}{3}\). Since these ratios differ, the sequence is not a geometric progression. Increasing terms or even terms do not make a sequence a GP. Exam tip: calculate ratios of consecutive terms and check whether they are equal.

The governing concept is the identification of the first term of a geometric progression. In a GP, the first term is denoted by a, and every later term is obtained by multiplying the preceding term by a fixed common ratio r. The displayed order is important: the first number written is the first term. Here the sequence starts with 12, so a = 12. The common ratio confirms the pattern because 36 ÷ 12 = 3, 108 ÷ 36 = 3, and 324 ÷ 108 = 3. Thus 3 is the common ratio, not the first term; 36 and 108 are later terms. Therefore, option B is correct.

The \(n\)th term of a geometric progression is \(T_n=ar^{n-1}\). Therefore, \(T_3=ar^2=2\times6^2=2\times36=72\). Here, \(36\) is only the value of \(r^2\); it must be multiplied by the first term, \(a=2\). Exam tip: for the third term, use \(ar^2\) directly.

Each term is multiplied by (\frac{1}{2}) so the next term is (16\times\frac{1}{2}=8). In exams identify fractional ratios in decreasing sequences.

The common ratio is (3) so the fifth term is (108\times3=324). In exams multiply the last known term by the ratio.

The ratio of (54) and (162) is (3) so the missing term is (54\div3=18). In exams apply the ratio backward too.

The direct answer is option A: \(a_n=5\cdot4^{n-1}\). The general term means a formula that gives any term when its position n is known. For a geometric progression, the formula is \(a_n=ar^{n-1}\), where a is the first term and r is the common ratio. Here a = 5 because the first term is 5. The ratio is 20 ÷ 5 = 4, also 80 ÷ 20 = 4. Substituting these values gives \(a_n=5\cdot4^{n-1}\). For example, n = 1 gives 5, n = 2 gives 20, and n = 3 gives 80. Option A has the correct first term and ratio. Option B, \(4\cdot5^{n-1}\), starts with 4 and uses the wrong ratio, so it does not give the sequence. Option C, 5n, is an arithmetic-style linear rule and gives 10 for n = 2, not 20. Option D, 20n, gives 20 for the first term instead of 5. Memory cue: GP formula is first term multiplied by ratio raised to one less than the position.

The nth term is \(a_n=6^n\). Substituting \(n=2\), we get \(a_2=6^2=36\). Note that \(216=6^3\), so it is \(a_3\), not \(a_2\). Exam tip: first substitute the required term number for \(n\) before evaluating the power.

The first term is (8) and the ratio is (5) so (a_n=8\cdot5^{n-1}). In exams use (a_n=ar^{n-1}).

In a geometric progression, the next term is found by multiplying the current term by the common ratio. Thus the governing formula for the second term is a₂ = a₁r. Substituting a₁ = 54 and r = 1/3 gives a₂ = 54 × 1/3 = 18. Therefore option B is correct. Because the ratio is less than 1, the sequence decreases from 54 to 18 rather than increasing. Option A would involve dividing 54 by 6, which is not the given rule. Option C would result from multiplying by 1/2, and option D incorrectly uses the reciprocal ratio 3. The calculation and the expected decrease agree.

The ratios (5\div2=\frac{5}{2}) and (10\div5=2) are not equal. In exams equal ratio is necessary for a geometric progression.

The common ratio of a geometric progression is the quotient obtained by dividing any term by the immediately preceding term. Using the first two terms, r = 45 ÷ 15 = 3. The result is verified by the later pairs: 135 ÷ 45 = 3 and 405 ÷ 135 = 3. Since the quotient remains constant, the sequence is indeed geometric and its common ratio is 3. Therefore option B is correct. If r were 2, the second term after 15 would be 30; if r were 5, it would be 75. Option D, 15, is the first term rather than the multiplier. Checking several consecutive pairs prevents confusing a term with the ratio.

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, not a ratio. Exam tip: compare ratios of consecutive terms.

The common ratio is (\frac{1}{2}) so the fifth term is (18\times\frac{1}{2}=9). In exams simplify fractions.

The terms are (5,15,45,135,405) so (405) is the fifth term. In exams list small terms in order to check.

Given \(a_n=6\cdot2^{n-1}\), for \(n=1\), \(a_1=6\cdot2^0=6\); for \(n=2\), \(a_2=6\cdot2^1=12\); and for \(n=3\), \(a_3=6\cdot2^2=24\). Therefore, the first three terms are \((6, 12, 24)\). Option B misses the initial coefficient 6. Exam tip: always substitute \(n=1\) first to check the first term.

The first term is (7) and the ratio is (4) so (a_n=7\cdot4^{n-1}). In exams match both the first term and the ratio.

The direct answer is option C: the missing term is 500. In a geometric progression, the same multiplier must connect every pair of consecutive terms. From 4 to 20, the multiplier is 20 ÷ 4 = 5. From 20 to 100, it is 100 ÷ 20 = 5, confirming the common ratio. Therefore the next term is 100 × 5 = 500. Option A, 300, would require multiplying 100 by 3, so it breaks the common ratio. Option B, 400, would require a multiplier of 4. Option C, 500, continues multiplication by 5 and is correct. Option D, 600, would require a multiplier of 6 and is also inconsistent. The dots after the blank do not change the calculation; they simply show that the sequence continues. Memory cue: divide a term by the previous term; if the quotient is fixed, multiply the last known term by that quotient.

The ratio of consecutive terms is (24\div72=\frac{1}{3}). In exams write the fractional ratio for a decreasing geometric progression.

After the fourth term (128) the fifth is (512) and the sixth is (2048). In exams include the first term while counting.

A sequence is a geometric progression when the quotient of every term and its preceding term is constant. Here, 42 ÷ 14 = 3, 126 ÷ 42 = 3, and 378 ÷ 126 = 3. Since the same ratio occurs throughout the displayed sequence, it is a geometric progression with common ratio 3. Therefore option A is correct. Option B mistakes the first term for the common ratio. Option C uses an irrelevant condition: the terms do not need to be equal; they need to have a constant multiplicative ratio. Option D is false because the terms increase, rather than decrease, as each term is tripled.

In a geometric progression, the first term is a, and each successive term is found by multiplying the preceding term by the common ratio r. Here, a = 4 and r = 7: 4, 4×7 = 28, 28×7 = 196, and 196×7 = 1372. Therefore, (4, 28, 196, 1372) is correct. Option A incorrectly treats 7 as the second term; it is the common ratio. Exam tip: write a first, then multiply each term by r.