Geometric Progression

The governing concept is a geometric progression, in which every term is obtained by multiplying the preceding term by the same constant ratio. Divide the second term by the first: \(r=(5/3)\div(5/6)=2\). The pattern is confirmed because \((10/3)\div(5/3)=2\) and \((20/3)\div(10/3)=2\). Therefore the next term is \(a_5=(20/3)\times2=40/3\). The general formula gives the same result: \(a_n=ar^{n-1}\), so \(a_5=(5/6)2^4=40/3\). Hence option B is correct. Option A does not continue the constant-ratio pattern, while C and D result from multiplying by an incorrect power or making an arithmetic error with the fractions.

The common ratio is (\frac{1}{3}), so the terms are (243,81,27,9,3,1). In a decreasing progression multiply by the ratio repeatedly.

In option A, consecutive ratios are \(-4/2=-2\), \(8/(-4)=-2\), and \((-16)/8=-2\). Hence it is a GP. In B, the ratios change. Exam tip: check successive ratios, not successive differences.

In (81,27,9,3,\ldots), (\frac{27}{81}=\frac{1}{3}). Divide consecutive terms to find the ratio.

A geometric progression has a fixed ratio between consecutive terms. Here \(r=25/5=5\), and the sequence is formed by multiplying by 5 at every step. Thus \(a_1=5\), \(a_2=25\), \(a_3=125\), and \(a_4=625\). One more multiplication gives \(a_5=625\times5=3125\), so the required number is the fifth term. Using the formula \(a_n=ar^{n-1}\), we obtain \(3125=5\times5^{n-1}=5^5\), hence \(n-1=4\) and \(n=5\). Therefore option B is correct. Option A stops at 625, whereas options C and D count one or two extra multiplications and therefore represent larger terms.

Using \(a_n=27\left(\frac{1}{3}\right)^{n-1}\), substitute \(n=4\): \(a_4=27\left(\frac{1}{3}\right)^{4-1}=27\left(\frac{1}{3}\right)^3=27\times\frac{1}{27}=1\). Therefore, the correct answer is 1. Option 3 would result from incorrectly using exponent 2, but here \(n-1=3\). Exam tip: write \(n-1\) separately before evaluating the power.

The common ratio is (\frac{16}{4}=4), so (x=16\cdot4=64). Apply the same ratio to each next term.

The first four terms are 14, 28, 56, and 112. Therefore, their sum is \(14+28+56+112=210\). Hence, 210 is the correct answer. Note that 224 is not the sum; it would be the next term of the progression. Exam tip: For a GP with only a few terms, direct addition is often the quickest method.

In \(2, 6, 18, 54\), the consecutive ratios are \(6/2=3\), \(18/6=3\), and \(54/18=3\), so it is a GP. In option C, the ratios are not constant. Exam tip: verify the ratio of every pair of consecutive terms.

The first term is (12) and the ratio is (3), so (a_n=12\cdot3^{n-1}). Use both first term and ratio in the general term.

The governing concept is the sum of the first n terms of a geometric progression. The first term is a = 3, the common ratio is r = 2, and n = 6. The terms are 3, 6, 12, 24, 48, and 96, whose direct sum is 3 + 6 + 12 + 24 + 48 + 96 = 189. The geometric-series formula gives the same result: for r ≠ 1, Sₙ = a(rⁿ − 1)/(r − 1). Hence S₆ = 3(2⁶ − 1)/(2 − 1) = 3(64 − 1) = 3 × 63 = 189. Therefore option A is correct. The other values can arise from an addition mistake, an incorrect power, or counting a different number of terms.

The governing rule for the nth term of a geometric progression is \(a_n=a_1r^{n-1}\). For the third term, the ratio is used twice because the sequence moves from the first term to the second and then from the second to the third. Hence \(a_3=30(1/3)^{3-1}=30(1/3)^2=30/9=10/3\). Directly, the first next term is \(30\times1/3=10\), and the following term is \(10\times1/3=10/3\), confirming the result. Therefore option A is correct. Option C is only the second term, option D is an unsimplified expression equal to the same value but is not the intended simplified choice, and option B does not follow the repeated ratio.

The common ratio is (\frac{40}{8}=5), so (x=40\cdot5=200). Multiply the second term by the ratio to get the third term.

This is a geometric progression because every term is obtained by multiplying the preceding term by the same number. Here, the common ratio is 4: 24 divided by 6 is 4, 96 divided by 24 is 4, and 384 divided by 96 is 4. Therefore, the terms grow as 6, 24, 96, 384, and so on.

Starting from the first term, the fifth term is obtained by multiplying by 4 four times. Thus the terms in order are first term 6, second term 24, third term 96, fourth term 384, and fifth term 1536. Hence 1536 is the fifth term, so option B is correct. The answer is not the fourth term because 384 is the fourth term.

The governing property is that terms of a geometric progression separated by several positions differ by the corresponding power of the common ratio. From the second term to the fifth term there are three steps, so \(a_5=a_2r^{5-2}=16r^3\). Substitution gives \(1024=16r^3\). Dividing by 16 yields \(r^3=64\), and the positive cube root is \(r=4\). A check gives \(16\times4^3=16\times64=1024\). Thus option C is correct. If \(r=2\), the result would be 128; if \(r=3\), it would be 432; and \(r=8\) would produce a much larger value. The positivity condition also makes the intended positive root explicit.

The defining concept is a geometric progression with a possibly negative common ratio. Dividing consecutive terms gives \(r=-15/5=-3\); this is confirmed because \(45/(-15)=-3\) and \((-135)/45=-3\). To obtain the fifth term, multiply the fourth term by the ratio: \(a_5=(-135)(-3)=405\). Using the formula gives the same result: \(a_5=5(-3)^4=5\times81=405\). The two negative factors produce a positive fifth term, which is why option A is correct. Option B ignores the sign change at the last multiplication, while C and D arise from using an incorrect magnitude or exponent.

The first term is (7), and multiplying by (4) each time gives (7,28,112,448). Match both (a) and (r).

The first three terms are 6, 18, and 54. Their product is \(6\times18\times54=108\times54=5832\). Hence, 5832 is correct. A value such as 3888 can result from an incorrect multiplication. Exam tip: multiply in steps—first find \(6\times18\), then multiply the result by 54.

The common ratio is (\frac{1}{3}), so the next term is (10\cdot\frac{1}{3}=\frac{10}{3}). Multiply by the ratio.

The formula gives the term corresponding to any positive integer value of \(n\). To identify the sequence, substitute the first few values of \(n\), beginning with \(n=1\). The exponent \(n-1\) is zero for the first term, so the first term is 8; every increase of one in \(n\) doubles the value.

For \(n=1,2,3,4\), the formula gives \(8\cdot2^0=8\), \(8\cdot2^1=16\), \(8\cdot2^2=32\), and \(8\cdot2^3=64\). Thus the sequence is \((8,16,32,64,\ldots)\), which matches option C. Option B starts with 2 and follows a different multiplier, while option D incorrectly begins with 16.