Geometric Progression

In the formula (a_n=7\cdot2^{n-1}) the common ratio is (2). In exams identify (r) in (ar^{n-1}).

The common ratio of this GP is \(r=42/14=3\). Here, \(378\) is the fourth term, so \(a_5=378\times 3=1134\). \(945\) is not correct because it is not obtained by multiplying the previous term by 3. Exam tip: count the given terms first, then multiply the last given term by the common ratio.

The \(n\)th term of a geometric progression is \(a_n=a_1r^{n-1}\). Hence, \(a_3=5\times4^{3-1}=5\times16=80\). Therefore, 80 is correct. The close distractor 20 is the second term, \(a_2=5\times4\), not the third term. Exam tip: when finding the \(n\)th term, use \(n-1\) as the exponent of \(r\).

In (81,27,9,3) each term is multiplied by (\frac{1}{3}). In exams write the ratio of a decreasing geometric progression as a fraction.

This is of the form (4\cdot2^{n-1}) so the tenth term is (4\cdot2^9=2048). In exams identify powers to solve quickly.

Given \(a_n=4^n\). Substituting \(n=3\), we get \(a_3=4^3=4\times4\times4=64\). The value \(16\) is \(4^2\), so it does not represent \(a_3\). Exam tip: when finding a term, substitute the given index correctly in the exponent.

The first term is (36) and the ratio is \(\frac{1}{2}\). In exams put the correct fractional ratio in \(ar^{n-1}\).

The square of the middle term is (8\times72=576) so the positive (x=24). In exams for three GP terms the square of the middle term equals the product of the outer terms.

The common ratio is (4) so the fifth term is (768\times4=3072). In exams multiply the last known term by the ratio.

For a geometric progression, two conditions must hold: the first term must equal the stated initial value, and every following term must be obtained by multiplying by the stated common ratio. Starting with 6 and repeatedly multiplying by 4 gives 6 × 4 = 24, 24 × 4 = 96, and 96 × 4 = 384. Therefore, option A satisfies both requirements exactly. Option B starts with 4 and has ratio 2. Option C has a constant difference of 4, so it is an arithmetic progression rather than a geometric one. Option D has ratio 4, but it starts with 24 instead of 6; it begins at the second term of the required progression. Hence only option A is correct.

In a geometric progression, the ratio of each term to the preceding term remains constant. In option A, 6/3 = 12/6 = 24/12 = 2, so it is a GP. Option B has a constant difference, so it is an AP. Exam tip: check ratios of consecutive terms.

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 terms are (9,27,81,243,729), so (729) is the fifth term. Form the terms in order to find the position.

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 \(b^2=ac\).