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.

In a geometric progression, the common ratio is the number by which each term is multiplied to obtain the next term. It is calculated by dividing the next term by the previous term. From the first two terms, \(r=\frac{15}{5}=3\). Checking further gives \(\frac{45}{15}=3\) and \(\frac{135}{45}=3\), so the same factor is used throughout the sequence.

Therefore option C, 3, is correct. Starting with 5, multiplication by 3 produces 15; another multiplication by 3 produces 45; and one more produces 135. A value of 2 would produce 10 after the first term, while 4 would produce 20, so neither matches. The value 5 is the first term itself, not the factor connecting consecutive terms. Division is the safest way to identify a GP ratio.

In a geometric progression, each term is obtained by multiplying the previous term by the common ratio. Thus, the second term is \(4\times5=20\), and the third term is \(20\times5=100\). Therefore, the correct answer is 100. The ratio must be applied twice to reach the third term. Exam tip: use \(a_n=a r^{n-1}\) for the \(n\)th term.

The common ratio is (\frac{48}{96}=\frac{1}{2}). In a decreasing geometric progression the ratio can be less than (1).

Given \(a_n=3\cdot2^{n-1}\), substitute \(n=6\): \(a_6=3\cdot2^{6-1}=3\cdot2^5=3\cdot32=96\). Hence, 96 is correct. The value 64 is only \(2^6\); it ignores the initial factor 3. Exam tip: Substitute the term number first, then simplify the exponent \(n-1\) carefully.

The direct answer is option B: 729 is the 5th term. A geometric progression is a sequence in which each term is obtained by multiplying the previous term by one fixed number, called the common ratio. Here, the ratio is 27 ÷ 9 = 3, and 81 ÷ 27 = 3. Starting from the first term, the terms are: 1st = 9, 2nd = 27, 3rd = 81, 4th = 243, and 5th = 243 × 3 = 729. Therefore, option B is correct. Option A says 4th, but the 4th term is 243, not 729. Option B says 5th and exactly matches the ordered list. Option C says 6th, but the 6th term would be 729 × 3 = 2187. Option D says 7th, which is even later and is also incorrect. The safest exam method is to write the terms in order and count from the first term. Memory cue: never count the gaps; count the actual terms, beginning with the first term as term 1.

The first term is (8), and multiplying by (3) each time gives (8,24,72,216). In a geometric progression each next term is formed by multiplication.

In a geometric progression, the common ratio is the quotient of consecutive terms. Here, \(r=\frac{-12}{4}=-3\). Checking further, \(\frac{36}{-12}=-3\) and \(\frac{-108}{36}=-3\). Hence, the correct answer is \(-3\). The closest distractor is \(3\), but a positive ratio would not make the signs alternate. Exam tip: divide any term by its preceding term to verify the common ratio.

The common ratio is (5), so the next term is (50\cdot5=250). Multiply the previous term by the ratio to get the next 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, \(x^2=4\times64=256\). Hence \(x=\pm16\), but the terms are stated to be positive, so \(x=16\). The nearby distractor \(8\) is incorrect because \(4,8,64\) does not have equal consecutive ratios. Exam tip: for three consecutive GP terms, use \(b^2=ac\).

The first four terms are 6, 18, 54, and 162. Their sum is \(6+18+54+162=240\). Hence, 240 is correct. A value such as 230 would result from an addition error or from omitting part of a term. Exam tip: for a GP with only a few terms, direct addition is often the quickest way to verify the sum.

In a geometric progression, the second term is \(a_2=a_1r\). Thus, \(21=a_1\times 3\), so \(a_1=21\div 3=7\). If 9 were the first term, the second term would be \(9\times 3=27\), not 21. Exam tip: To move back one term in a GP, divide by the common ratio.

The governing pattern is geometric because each term is obtained by multiplying the preceding term by 2. Indeed, (3/2) ÷ (3/4) = 2, 3 ÷ (3/2) = 2, and 6 ÷ 3 = 2. Thus the first term is 3/4 and the common ratio is 2. Continuing the sequence, the fifth term is 6 × 2 = 12 and the sixth term is 12 × 2 = 24. Therefore, option B is correct. Option A is only the fifth term. Options C and D do not result from continuing the same doubling rule. The formula confirms the calculation: a₆ = (3/4) × 2⁵ = (3/4) × 32 = 24.

The common ratio is (\frac{1}{2}), so the terms are (128,64,32,16,8,4). In a decreasing progression multiply by the ratio repeatedly.

The nth term of a geometric progression is \(a_n=ar^{n-1}\). Thus, \(a_4=2\times6^{4-1}=2\times6^3=2\times216=432\). Therefore, 432 is the correct option. \(216\) is only the value of \(6^3\); it still needs to be multiplied by the first term, \(a=2\). Exam tip: for the nth term, the exponent of \(r\) is \(n-1\), not \(n\).

In (100,20,4,\frac{4}{5},\ldots), (\frac{20}{100}=\frac{1}{5}). Divide consecutive terms to find the ratio.

Direct answer: 486 is the sixth term, so option C is correct. Each term is three times the preceding term, so this is a geometric progression with first term 2 and ratio 3. Continue the sequence carefully: first 2, second 6, third 18, fourth 54, fifth 162, and sixth 486. The general-term check is aₙ = 2 × 3ⁿ⁻¹. For n = 6, a₆ = 2 × 3⁵ = 2 × 243 = 486. Option A is wrong because the fourth term is 54. Option B is wrong because the fifth term is 162. Option C matches the sixth term exactly. Option D is wrong because the seventh term would be 486 × 3 = 1458. The important skill is to count the first given term as term 1; do not begin counting after it. Memory cue: write the term number above each value before deciding.

Given \(a_n=16\left(\frac{1}{2}\right)^{n-1}\). Substituting \(n=4\), \(a_4=16\left(\frac{1}{2}\right)^{4-1}=16\left(\frac{1}{2}\right)^3=16\times\frac{1}{8}=2\). Hence, the correct answer is \(2\). The value \(4\) would result from using exponent \(2\), but here \(n-1=3\). Exam tip: after substituting the term number, check the exponent \(n-1\) separately.

The direct answer is option C: x = 48. In a geometric progression, the quotient of every term and the term before it is the same. From the first two terms, the common ratio is 12 ÷ 3 = 4. Thus the third term must be 12 × 4 = 48. Checking further, the fourth term becomes 48 × 4 = 192, exactly as given. Option A, 36, would not continue the ratio 4 because 36 ÷ 12 = 3. Option B, 42, gives 42 ÷ 12 = 3.5, so it also breaks the fixed-ratio rule. Option C, 48, gives 48 ÷ 12 = 4 and then 192 ÷ 48 = 4, so it works. Option D, 64, gives a ratio of 64 ÷ 12, not 4, and would produce 256 rather than 192 as the next term. The important idea is to find the ratio from two known consecutive terms and use it again. Memory cue: in a GP, multiply by the same number, not by changing numbers.

The first four terms are 12, 24, 48, and 96. Therefore, their sum is \(12+24+48+96=180\). The value 192 may result from an incorrect addition, since the fourth term is 96. Exam tip: When only a few terms are given, add them directly and check the total.

In a geometric progression, the third term is \(a_3=a_1r^2\). Thus, \(45=a_1\times 3^2=9a_1\), so \(a_1=45/9=5\). Option 15 is the second term \(a_2\), since \(5\times3=15\). Exam tip: use \(a_n=a_1r^{n-1}\) for the nth term of a GP.

The first term is (10) and the ratio is (2), so (a_n=10\cdot2^{n-1}). Use both first term and ratio in the general term.

This is a geometric progression with first term 2 and common ratio 3. Its first 5 terms are 2, 6, 18, 54, and 162. Their sum is 2+6+18+54+162=242. Therefore, 242 is the correct answer. Option 200 is incorrect because it is not the sum of these five terms. Exam tip: You may also use \(S_n=\frac{a(r^n-1)}{r-1}\) when \(r\ne1\).