Geometric Progression

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.

The common ratio is (\frac{1}{3}), so the next term is (8\cdot\frac{1}{3}=\frac{8}{3}). Multiply by the ratio.

The formula \\(a_n=6\\cdot3^{n-1}\\) defines a geometric progression. Substitute consecutive values of n to obtain its beginning. For \\(n=1\\), \\(a_1=6\\cdot3^0=6\\). For \\(n=2\\), \\(a_2=6\\cdot3=18\\); for \\(n=3\\), \\(a_3=6\\cdot9=54\\); and for \\(n=4\\), \\(a_4=6\\cdot27=162\\).

Therefore, the sequence begins \\(6,18,54,162,\\ldots\\), exactly as shown in option C. Each term is three times the preceding term. Option A is an arithmetic sequence, option B adds 3 each time, and option D starts with 18 rather than 6. Hence option C is the only choice matching the given rule.

In a geometric progression, the ratio of consecutive terms is constant. Here, \(r=\frac{54}{18}=3\). Therefore, \(\frac{18}{x}=3\), giving \(x=\frac{18}{3}=6\). If 9 were used, the ratios would be \(\frac{18}{9}=2\) and \(\frac{54}{18}=3\), which are not equal. Exam tip: for three consecutive GP terms \(a,b,c\), use \(b^2=ac\).

In this geometric progression, the first term is \(a=5\) and the common ratio is \(r=\frac{20}{5}=4\). Using \(a_n=ar^{n-1}\), \(a_5=5\times4^{5-1}=5\times4^4=1280\). Getting 2560 would mean multiplying by the ratio 4 one extra time. Exam tip: in the \(n\)th-term formula, the exponent is always \(n-1\).

In (3,12,48,192,\ldots), each term is multiplied by (4). For the ratio divide the next term by the previous term.

In a geometric progression, the \(n\)th term is \(a_n=a_1r^{n-1}\). Therefore, \(a_5=\frac{5}{8}\times2^{5-1}=\frac{5}{8}\times16=10\). Option 8 can result from an incorrect use of the exponent or multiplication. Exam tip: in \(a_n\), the exponent of the common ratio is always \(n-1\), not \(n\).

The first term of this GP is 3 and its common ratio is 2. Its first five terms are 3, 6, 12, 24, and 48, whose sum is 93. Using the formula, \(S_5=\frac{3(2^5-1)}{2-1}=93\). The value 90 is the sum of only the first four terms. Exam tip: List the required number of terms before adding to avoid missing the last term.

The defining rule of a geometric progression is an = an-1 × r. The first term is already given as a1 = 4, and each following term must be multiplied by -3. Thus a2 = 4 × (-3) = -12, a3 = (-12) × (-3) = 36, and a4 = 36 × (-3) = -108. The first four terms are therefore 4, -12, 36, -108, so option B is correct. The alternating signs are essential because the common ratio is negative. Option A ignores the negative sign, option C treats -3 as the second term rather than as the multiplier, and option D incorrectly changes the sign of the first term.

For consecutive terms of a geometric progression, the square of the middle term equals the product of its neighbouring terms. Thus, \(x^2=9\times49=441\). Hence \(x=\pm21\), but the progression is stated to be positive, so \(x=21\). For example, \(28^2\ne9\times49\), so 28 cannot be the middle term. Exam tip: for three consecutive GP terms \(a,b,c\), use \(b^2=ac\) directly.

The first term is (11) and the ratio is (3), so (a_n=11\cdot3^{n-1}). Use the first term and ratio in the general term.

In a geometric progression, each term is obtained by multiplying the previous term by the same common ratio. Moving from the second term to the fifth term involves three equal steps, so the multiplier is used three times. This is why the position difference becomes the exponent in the formula.

Using the given values, \\(a_5=a_2r^3\\). Therefore, \\(810=30r^3\\), and dividing by 30 gives \\(r^3=27\\). Since the ratio is stated to be positive, the suitable real value is \\(r=3\\). Thus option B follows. A value such as 9 is not correct because the ratio is cubed and must be checked in the original relation.

Direct answer: a₅ = 1024, so option C is correct. The common ratio is r = 16/4 = 4, and 64/16 = 4 confirms that every term is multiplied by 4. Using aₙ = a₁rⁿ⁻¹, we get a₅ = 4 × 4⁴ = 4 × 256 = 1024. A direct continuation is also clear: the fourth term is 64 × 4 = 256, and the fifth is 256 × 4 = 1024. Option A does not follow the repeated multiplication by 4. Option B is not a term of this progression. Option C is exactly the fifth term. Option D is twice the required value and would require an incorrect extra factor of 2. Remember that the first term 4 already supplies one factor of 4; the ratio is applied only four additional times to reach the fifth term.

(a_3=a_1r^2), so (75=3r^2) and (r^2=25), hence positive (r=5). When positive ratio is asked take the positive root.

Each term is (\frac{1}{3}) of the previous term, so (r=\frac{1}{3}). To check the ratio divide the next term by the previous term.

This is a geometric progression with first term 2 and common ratio 4. Hence, \(a_n=2\times4^{n-1}\). From \(2\times4^{n-1}=512\), we get \(4^{n-1}=256=4^4\). Therefore, \(n-1=4\), so \(n=5\). At \(n=4\), the term is 128, not 512. Exam tip: use \(a_n=ar^{n-1}\) to find the position of a term in a GP.

In the given GP, the second term is \(a_2=12\) and the fourth term is \(a_4=108\). Therefore, \(a_2+a_4=12+108=120\). The value 112 comes from adding \(a_1+a_4=4+108\), so it is not correct here. Exam tip: count terms by taking the first term as \(a_1\).

The governing idea is that each term of a geometric progression is obtained by multiplying the previous term by r. With a = 24 and r = 1/2, the first four terms are 24, 12, 6, and 3. Their sum is 24 + 12 + 6 + 3 = 45, so option D is correct. The finite geometric-sum formula gives the same result: S4 = a(1-r^4)/(1-r) = 24[1-(1/2)^4]/(1/2) = 45. The distractors 40, 42, and 44 result from omitting a term, making an arithmetic error, or using the ratio incorrectly. Because the ratio is less than one, the terms decrease but must all still be included.

In a geometric progression, each term is obtained by multiplying the preceding term by the common ratio. If a term is already known, we can reach a later term by multiplying by the ratio once for every step between their positions. The exponent therefore represents the number of steps, not simply the later term number.

From the second term to the fifth term there are three steps: second to third, third to fourth, and fourth to fifth. Thus, using the given ratio, \(a_5=a_2r^{5-2}=18\times2^3\). Since \(2^3=8\), the value is \(18\times8=144\). Therefore, option C is correct. Option A would use too few multiplications.