Geometric Progression

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 common ratio is (\frac{12}{3}=4), so (x=12\cdot4=48). Apply the same ratio to each next term.

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 \(r\ne1\).

Direct answer: a₄ = 5/2, so option B is correct. In a geometric progression, aₙ = a₁rⁿ⁻¹. The exponent is n − 1 because moving from the first term to the second uses the ratio once, to the third twice, and to the fourth three times. Substitute the values: a₄ = 20(1/2)³ = 20 × 1/8 = 20/8 = 5/2. A direct sequence check gives 20, 10, 5, 5/2. Option A is the third term, reached after halving twice. Option B is the fourth term and follows the rule exactly. Option C is the second term, not the fourth. Option D does not result from repeatedly multiplying by 1/2. The common error is using the exponent 4 instead of 3; remember that the first term needs zero ratio applications.

The common ratio is (\frac{30}{6}=5), so (x=30\cdot5=150). Multiply the second term by the ratio to get the third term.

Direct answer: 1875 is the fifth term, so option B is correct. The sequence is geometric because each term is multiplied by 5: 15/3 = 5, 75/15 = 5, and 375/75 = 5. Therefore the next term is 375 × 5 = 1875. Counting gives first 3, second 15, third 75, fourth 375, and fifth 1875. The formula confirms this: aₙ = 3 × 5ⁿ⁻¹. Setting it equal to 1875 gives 5ⁿ⁻¹ = 625 = 5⁴, so n − 1 = 4 and n = 5. Option A is the fourth term, 375. Option B is the next term and is correct. Option C would be 1875 × 5 = 9375, not 1875. Option D would be 46875 after one more multiplication by 5. The useful check is to verify the ratio before using a formula.

In a geometric progression, terms at positions that are three places apart are related by three powers of the common ratio. Since \\(a_2=12\\) and \\(a_5=324\\), moving from the second term to the fifth term involves three steps. Therefore, \\(a_5=a_2r^3\\), so \\(324=12r^3\\).

Dividing by 12 gives \\(r^3=27\\). The real cube root of 27 is 3, and the question states that the ratio is positive, so \\(r=3\\). Hence option B is correct. The positive condition removes any concern about a negative cube-root alternative; in any case, a negative ratio would not satisfy the stated condition.

A geometric progression has a constant multiplier between consecutive terms. Here the common ratio is r = −4/2 = −2, and the later ratios confirm the same value: 8/(−4) = −2 and (−16)/8 = −2. Starting with 2, the fifth term is (−16)(−2) = 32, and the sixth term is 32(−2) = −64. Therefore option C is correct. The negative ratio causes the signs to alternate, while its magnitude 2 doubles the absolute value at each step. Option A is the fifth term, option B has the wrong sign, and option D has neither the correct magnitude nor the correct position. The formula aₙ = 2(−2)ⁿ⁻¹ also gives a₆ = 2(−2)⁵ = −64.

The first term is (12), and multiplying by (2) each time gives (12,24,48,96). Match both (a) and (r).

The first three terms are 4, 20, and 100. Their product is \(4\times20\times100=80\times100=8000\). An option such as 6000 can result from an incorrect multiplication. Exam tip: multiply the first two terms first, then multiply by the third term.