Geometric Progression

This is a geometric progression with first term 1 and common ratio 4. Its nth term is \(a_n=4^{n-1}\). Since \(4^{n-1}=256=4^4\), we get \(n-1=4\), so \(n=5\). Although 4 is the exponent, it is not the term number because the first term is \(4^0\). Exam tip: When a sequence starts with 1, add 1 to the exponent to obtain the term position.

In the given geometric progression, the third term is \(a_3=18\) and the fourth term is \(a_4=54\). Therefore, \(a_3+a_4=18+54=72\). A value such as 66 results from misreading one of the terms; the correct sum of the two given terms is 72. Exam tip: Count term positions carefully, starting with the first term.

Direct answer: the sum is 30, so option C is correct. A geometric progression is formed by multiplying each term by a fixed ratio. Starting with a = 16 and r = 1/2, the first four terms are 16, 8, 4, and 2. Add them carefully: 16 + 8 = 24, 24 + 4 = 28, and 28 + 2 = 30. Thus S₄ = 30. The finite geometric-sum formula gives the same check: Sₙ = a(1 − rⁿ)/(1 − r), so S₄ = 16[1 − (1/2)⁴]/(1 − 1/2) = 30. Option A is the partial sum 16 + 8 and leaves out two terms. Option B is the partial sum through the third term and leaves out 2. Option C includes all four required terms. Option D is larger than the correct total and may come from treating the terms incorrectly. The main warning is to count exactly four terms, including the first term.

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 first term is (2) and the ratio is (4) so (a_n=2\cdot4^{n-1}). In exams use (a_n=ar^{n-1}).

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 \(4-1=3\).