Geometric Progression

The governing property is that consecutive terms of a geometric progression have a constant ratio. For three consecutive terms, the square of the middle term equals the product of its neighbors, so x^2 = 5 × 80 = 400. Because all terms are positive, x = 20 rather than −20. The common ratio is therefore r = 20/5 = 4. Multiplying the third term by the same ratio gives y = 80 × 4 = 320. Thus option C is correct. The sequence 5, 20, 80, 320 confirms that the ratio is 4 in every step. The other options fail to preserve a constant ratio between the third and fourth terms or do not agree with the positive middle term.

The governing concept is the finite sum of a geometric progression, Sₙ = a₁(1 − rⁿ)/(1 − r), which is valid because r ≠ 1. Substituting a₁ = 10, r = −2, and n = 5 gives S₅ = 10[1 − (−2)⁵]/[1 − (−2)] = 10(1 + 32)/3 = 330/3 = 110. Direct expansion confirms the result: the five terms are 10, −20, 40, −80, and 160, so their sum is 10 − 20 + 40 − 80 + 160 = 110. Hence option C is correct. The negative ratio makes signs alternate, but the odd power (−2)⁵ is negative, which changes the numerator to 33. Treating the ratio as positive or ignoring that sign produces distractor values.

For three consecutive terms of a geometric progression, the square of the middle term equals the product of the first and third terms. Therefore (5x)^2 = x × 125. Expanding gives 25x^2 = 125x, so 25x(x − 5) = 0. The algebraic solutions are x = 0 and x = 5, but the question asks for positive x, so only x = 5 is acceptable. Checking directly, the terms become 5, 25 and 125, and the ratio is 5 from one term to the next. Thus option C is correct. The zero solution must be rejected because zero is neither positive nor suitable for the intended nonzero geometric progression.

This question applies the nth-term rule of a geometric progression, aₙ = a₁rⁿ⁻¹. Since a₁ = 10 and r = 3, the fourth term is a₄ = 10×3³ = 10×27 = 270. The sixth term is a₆ = 10×3⁵ = 10×243 = 2430. Therefore, the required difference is a₆ − a₄ = 2430 − 270 = 2160, so option A is correct. Careful indexing is essential: the fourth term uses the third power of 3, and the sixth term uses the fifth power. Writing only 2430 gives a₆ rather than the requested difference. The other choices can arise from using an incorrect exponent, multiplying the terms instead of subtracting them, or making an arithmetic error. Listing the sequence, 10, 30, 90, 270, 810, 2430, also verifies both indexed values and the final subtraction.

In a geometric progression, moving two positions forward multiplies a term by r^2. Thus a_5 = a_3r^2, so 360 = 40r^2 and r^2 = 9. Since r is positive, r = 3. Moving from the fifth term to the seventh term also involves two steps, so a_7 = a_5r^2 = 360 × 9 = 3240. Therefore option C is correct. An alternative check uses a_7 = a_3r^4 = 40 × 3^4 = 40 × 81 = 3240. The other options do not result when the same two-step multiplier is applied consistently, and using r rather than r^2 would incorrectly ignore one of the two transitions.

The common ratio is found by dividing consecutive terms: r = -12/3 = -4, and the later ratios confirm it. The nth-term formula is a_n = a_1r^(n-1). Thus a_6 = 3(-4)^5. Because the exponent 5 is odd, (-4)^5 = -1024, and multiplying by 3 gives a_6 = -3072. Therefore option D is correct. The sign alternates because the ratio is negative; after the fifth multiplication the result remains negative. Options A and C have positive signs, and option B has the wrong magnitude, usually caused by using one fewer factor of -4.

The governing concept is the finite sum formula for a geometric progression: Sₙ = a(rⁿ − 1)/(r − 1) when r ≠ 1. Here a = 3, r = 3, and n = 5. Substitution gives S₅ = 3(3⁵ − 1)/(3 − 1) = 3(243 − 1)/2 = 3 × 242/2 = 363. Direct verification gives the five terms 3, 9, 27, 81, and 243, whose sum is also 363. Therefore the statement is true and option A is correct. Option B is too small and option C is too large. Option D gives only the fifth term, 243, rather than the sum of all five terms, so it does not correctly justify the statement even though it uses a genuine term of the progression.

The governing idea is the finite-sum formula for a geometric progression. The first term is a = 1, the common ratio is r = 4, and the number of terms is n = 6. Since r ≠ 1, Sₙ = a(rⁿ − 1)/(r − 1). Therefore S₆ = 1(4⁶ − 1)/(4 − 1) = (4096 − 1)/3 = 4095/3 = 1365. A direct calculation confirms this result: 1 + 4 + 16 + 64 + 256 + 1024 = 1365. Thus option A is correct. Options B, C, and D are nearby distractors likely caused by failing to subtract 1, using the wrong number of terms, or making an arithmetic error while dividing 4095 by 3.

The relevant concept is the sum of the first four terms of a geometric progression. Since r = 4 is not 1, use Sₙ = a(rⁿ − 1)/(r − 1). With a = 8 and n = 4, S₄ = 8(4⁴ − 1)/(4 − 1) = 8(256 − 1)/3 = 8 × 255/3 = 8 × 85 = 680. A direct check gives the terms 8, 32, 128, and 512; adding them yields 8 + 32 + 128 + 512 = 680. Consequently, the stated value is true and option A is correct. Options B, C, and D are incorrect totals. They may reflect a wrong power, omission or alteration of a term, or a simple arithmetic error, but none agrees with both the formula and direct addition.

The governing concept is the general term of a geometric progression, aₙ = arⁿ⁻¹. The first term is a = 3 and the common ratio is r = 12/3 = 4. Therefore a₅ = 3 × 4⁴ = 3 × 256 = 768, while a₆ = 3 × 4⁵ = 3 × 1024 = 3072. Adding the two requested terms gives a₅ + a₆ = 768 + 3072 = 3840, so option B is correct. Option A is only the sixth term and does not include a₅. Options C and D can result from using an incorrect exponent or adding incorrectly. Computing both terms separately is the safest approach because the question asks for a sum of two terms, not for a single term or for the sum of all terms.

This is an explicit general rule for a geometric progression. Substitute n = 4 and n = 5 separately into aₙ = 6 × 5ⁿ⁻¹. We get a₄ = 6 × 5³ = 6 × 125 = 750 and a₅ = 6 × 5⁴ = 6 × 625 = 3750. Therefore a₄ + a₅ = 750 + 3750 = 4500. Option A is correct. Option B could result from an inaccurate addition, option C from using a wrong power or term value, and option D is another overestimate. The exponent n − 1 is important: the first term uses 5⁰, so replacing it with 5ⁿ would shift every term and produce an incorrect result.

Direct answer: the sixth term is 2, so B is correct. Each term is half of the preceding term: 32 ÷ 64 = 1/2 and 16 ÷ 32 = 1/2. Thus the common ratio is r = 1/2. Counting carefully gives a₁ = 64, a₂ = 32, a₃ = 16, a₄ = 8, a₅ = 4, and a₆ = 2. The formula confirms the list: aₙ = arⁿ⁻¹, so a₆ = 64(1/2)⁵ = 64/32 = 2. D, 8, is the fourth term, and C, 4, is the fifth term. A, 1, would be the seventh term after one more division by 2. The common mistake is to count the number of divisions rather than the term number: reaching the sixth term requires five multiplications by 1/2 because the first term is already given. Memory cue: term number n means n−1 ratio steps from the first term.