Geometric Progression

The common ratio is \(-\frac{1}{2}\), so the next term is \((-6)\left(-\frac{1}{2}\right)=3\). With a negative ratio the signs alternate.

Given \(a_n=7\cdot2^{n-1}\), \(a_4=7\cdot2^3=56\) and \(a_5=7\cdot2^4=112\). Therefore, \(a_4+a_5=56+112=168\). Option 112 is only the value of \(a_5\), not the required sum. Exam tip: substitute the value of \(n\) carefully in the formula for each term.

Direct answer: 3072 is the 6th term, so A is correct. First identify the pattern: 12 ÷ 3 = 4, 48 ÷ 12 = 4, and 192 ÷ 48 = 4. Thus this is a geometric progression with first term a = 3 and common ratio r = 4. The nth-term formula is aₙ = arⁿ⁻¹. Put the required value into the formula: 3 × 4ⁿ⁻¹ = 3072. Dividing by 3 gives 4ⁿ⁻¹ = 1024. Since 1024 = 4⁵, n − 1 = 5, so n = 6. A direct check gives 3 × 4⁵ = 3 × 1024 = 3072. B is wrong because the 7th term is 3072 × 4 = 12288. C and D are even later terms and therefore are still larger. Memory cue: in a GP, count powers from the first term; the first term has power zero, not power one.

The governing concept is the sum formula for the first n terms of a geometric progression when r ≠ 1: Sₙ = a(rⁿ − 1)/(r − 1). Substituting a = 2, r = 3 and n = 5 gives S₅ = 2(3⁵ − 1)/(3 − 1). Since 3⁵ = 243, this becomes 2(242)/2 = 242. Therefore the statement claiming S₅ = 242 is true, so option A is correct. Direct checking gives the same result: the first five terms are 2, 6, 18, 54 and 162, whose sum is 242. Option B is short by 2, option C is greater by 2, and option D is exactly half the correct sum.

This uses the finite geometric-series sum formula. For a GP with first term a and ratio r ≠ 1, Sₙ = a(rⁿ − 1)/(r − 1). In this sequence, a = 1, r = 3 and n = 7. Thus S₇ = 1(3⁷ − 1)/(3 − 1) = (2187 − 1)/2 = 2186/2 = 1093. Therefore option A is correct. A useful verification is to add the terms directly: 1 + 3 + 9 + 27 + 81 + 243 + 729 = 1093. Option B is one less than the correct total and may result from subtracting incorrectly; option C has no valid calculation behind it, while option D is only the seventh term, not the sum of all seven terms.

The governing idea is the nth-term formula aₙ = arⁿ⁻¹. In this progression, a = 1 and r = 4. To find which term equals 1024, write 1 × 4ⁿ⁻¹ = 1024. Since 1024 = 4⁵, we get n − 1 = 5 and therefore n = 6. The terms confirm this: 1, 4, 16, 64, 256, 1024, so there are six terms through 1024. Hence option B is correct. Option A would stop at 256, the fifth term. Option C would include the next term, 4096, and option D would go even farther. The phrase “up to 1024” means the count ends when 1024 is reached.

The governing concept is the finite sum of a geometric progression. The first five terms can be generated by repeatedly multiplying by r = 1/3: 81, 27, 9, 3 and 1. Adding them gives 81 + 27 + 9 + 3 + 1 = 121, so option B is correct. The formula confirms this result: S₅ = a(1 − r⁵)/(1 − r) = 81[1 − (1/3)⁵]/(1 − 1/3) = 121. Because the ratio is less than one, the terms decrease, but all five terms must still be included. Option A omits one unit, while options C and D are larger than the correctly calculated sum. Direct addition is especially efficient here because the number of terms is small.

The governing concept is an explicit formula for the terms of a geometric progression. Substitute n = 4 into aₙ = 3 × 4ⁿ⁻¹: a₄ = 3 × 4³ = 3 × 64 = 192. Next, substitute n = 5: a₅ = 3 × 4⁴ = 3 × 256 = 768. Therefore a₄ + a₅ = 192 + 768 = 960, so option A is correct. A quick check is that the ratio between consecutive terms is 4, and 768 is four times 192. Option B is larger than the actual sum, option C does not result from the stated formula, and option D is also too large. The exponent must be n − 1; using n instead would shift both terms and produce an incorrect result.

The governing concept is the finite sum of a geometric progression. With first term 32 and ratio one-half, successive terms are found by halving: \(32,16,8,4\). Adding the four required terms gives \(32+16+8+4=60\). The formula \(S_n=a(1-r^n)/(1-r)\) provides an independent check: \(S_4=32[1-(1/2)^4]/[1-1/2]=32(15/16)/(1/2)=60\). Hence option D is correct. The distractors 50, 54, and 58 do not equal the complete sum; they can arise from omitting a term, using an incorrect power, or making an addition error. The direct method and formula agree exactly.

Use the geometric-progression relation between separated terms. From the second term to the sixth term there are four ratio steps, so \(a_6=a_2r^4\). Substitution gives \(160=10r^4\), hence \(r^4=16\). Since the ratio is positive, \(r=2\), not the negative fourth-root possibility. Now the second term satisfies \(a_2=a_1r\), so \(10=2a_1\), which gives \(a_1=5\). Therefore option D is correct. Verification is immediate: the sequence begins \(5,10,20,40,80,160\). The other choices do not produce both the stated second term and the stated sixth term under the same positive ratio.

The governing concept is the relationship between terms of a geometric progression: a_m = a_k r^(m−k). Moving from the second term to the fifth term requires three multiplications by r, so a₅ = a₂r³. Substituting the given values gives 405 = 15r³, hence r³ = 27. Since r is specified as positive, r = 3. Now use a₂ = a₁r: 15 = 3a₁, so a₁ = 5. Therefore option C is correct. The value 3 is the common ratio, not the first term. Values 4 and 6 do not produce both a₂ = 15 and a₅ = 405 with one positive common ratio, so they are not consistent with the progression.

The governing concept is the finite sum of a geometric progression, while the negative ratio requires careful sign handling. Starting with a₁ = 4 and multiplying successively by −2 gives the first five terms: 4, −8, 16, −32, and 64. Their signed sum is 4 − 8 + 16 − 32 + 64 = 44. The formula confirms this result: S₅ = a₁(1−r⁵)/(1−r) = 4[1−(−2)⁵]/[1−(−2)] = 4(33)/3 = 44. Thus option D is correct. Adding absolute values would incorrectly give a different result because the signs alternate. Options A, B, and C can arise from ignoring a negative sign, omitting a term, or making an arithmetic error.

For three consecutive terms p, q, and s of a geometric progression, the middle-term property is q² = ps. Applying this property to x, 3x, and 27 gives (3x)² = x·27. Thus 9x² = 27x, so 9x(x−3) = 0. The algebraic solutions are x = 0 and x = 3. Because the question specifically asks for the positive value, x = 3 must be selected. Option C is correct. Substitution verifies the result: the terms become 3, 9, and 27, and both consecutive ratios are 9/3 = 3 and 27/9 = 3. Zero is not positive, and 1, 2, and 9 fail the middle-term relation or do not produce equal consecutive ratios.

The governing concept is the nth-term formula for a geometric progression: aₙ = a₁rⁿ⁻¹. The exponent is n − 1 because the first term has undergone zero multiplications by the common ratio. With a₁ = 6 and r = 3, the fourth term is a₄ = 6×3³ = 6×27 = 162, and the sixth term is a₆ = 6×3⁵ = 6×243 = 1458. Hence a₆ − a₄ = 1458 − 162 = 1296, so option C is correct. Option D is only the value of a₆, not the requested difference. Options A and B do not result from correctly finding both indexed terms and subtracting the fourth from the sixth. Checking the sequence gives 6, 18, 54, 162, 486, 1458, which independently confirms that the required difference is 1296.

The governing concept is the general term of a geometric progression, aₙ = a₁rⁿ⁻¹, together with multiplication of the required terms. Starting from a₁ = 1 and multiplying successively by r = 4 gives a₁ = 1, a₂ = 4, a₃ = 16, and a₄ = 64. Therefore the product is 1 × 4 × 16 × 64. Since 4 × 16 = 64 and 64 × 64 = 4096, the correct answer is 4096, or option C. An efficient check is a₁⁴r^(0+1+2+3) = 1⁴ × 4⁶ = 4096. The other choices can arise from using the wrong number of powers, omitting a term, or making an arithmetic error; none equals the product of all four specified terms.

The governing idea is that equal index gaps in a geometric progression correspond to equal multiplicative factors. From a₂ to a₄ there are two steps, so a₄ = a₂r². Substituting gives 80 = 20r², hence r² = 4. Since r is positive, r = 2. From a₄ to a₆ there are again two steps, so the same factor r² = 4 applies. Therefore a₆ = a₄r² = 80 × 4 = 320, making option C correct. Option A uses only one multiplication by r, option B does not follow any valid two-step factor, and option D uses an excessive factor. The positive-ratio condition removes the negative value r = −2.

The governing concept is the nth-term formula for a geometric progression: aₙ = a₁rⁿ⁻¹. In this sequence, a₁ = 1 and the common ratio is r = −3, because each term is obtained by multiplying the preceding term by −3. Hence a₇ = 1 × (−3)⁶. The exponent is even, so the sign is positive, and 3⁶ = 729. Thus a₇ = 729, making option C correct. The sign pattern independently confirms this result: the first, third, fifth, and seventh terms are positive, while the even-position terms are negative. The negative options incorrectly retain a negative sign for an even power. The value 243 comes from using an incorrect exponent or incomplete power calculation, not from the seventh term.

The governing concept is the nth term of a geometric progression. In the sequence 7, 14, 28, 56, the first term is a = 7 and each term is multiplied by the common ratio r = 2. The nth-term formula is a_n = a × r^(n−1). Therefore, the twelfth term is a_12 = 7 × 2^11 = 7 × 2048 = 14,336. Hence option B is correct. A represents an earlier power calculation, while C and D are respectively twice and four times the correct value; they can result from using the wrong exponent or continuing the doubling one or two extra steps. The index n−1 is essential.

The governing concept is the finite sum of a geometric progression. The first term is a = 3, the common ratio is r = 3, and six terms are required. Because r ≠ 1, use Sₙ = a(rⁿ − 1)/(r − 1). Substitution gives S₆ = 3(3⁶ − 1)/(3 − 1) = 3(729 − 1)/2 = 3 × 728/2 = 1092. A direct verification lists the terms as 3, 9, 27, 81, 243, and 729; adding them gives 3 + 9 + 27 + 81 + 243 + 729 = 1092. Therefore option A is correct. The nearby alternatives 1093, 1095, and 1098 are not valid sums and could result from an addition slip, an incorrect power, or an arbitrary adjustment.

The governing concept is the finite sum of the first n terms of a geometric progression. Here a = 4, r = 4, and n = 4. Since r ≠ 1, Sₙ = a(rⁿ − 1)/(r − 1). Thus S₄ = 4(4⁴ − 1)/(4 − 1) = 4(256 − 1)/3 = 4 × 255/3 = 340. Direct listing gives the same result: the four terms are 4, 16, 64, and 256, and their sum is 4 + 16 + 64 + 256 = 340. Therefore the stated value is true and option A is correct. Option D confuses the final term, 256, with the sum. Options B and C agree with neither the formula nor direct addition.

The governing concept is the sum formula for a geometric progression. From 2, 8, 32 and 128, the first term is a = 2 and the common ratio is r = 4. For seven terms, use S_n = a(r^n − 1)/(r − 1). Hence S_7 = 2(4^7 − 1)/(4 − 1). Since 4^7 = 16,384, this becomes 2(16,383)/3 = 32,766/3 = 10,922. Therefore option B is correct. Direct addition of 2 + 8 + 32 + 128 + 512 + 2048 + 8192 also gives 10,922. The adjacent alternatives differ by small amounts and reflect arithmetic or power-calculation errors, not a valid geometric sum.

The governing concept is the general term of a geometric progression. In 2, 6, 18, the first term is a = 2 and the common ratio is r = 3. Using a_n = a × r^(n−1), we obtain a_6 = 2 × 3^5 = 2 × 243 = 486 and a_7 = 2 × 3^6 = 2 × 729 = 1458. Their sum is a_6 + a_7 = 486 + 1458 = 1944. Thus option A is correct. Option B may result from using an incorrect power, while C and D are larger values caused by extending the progression or adding terms incorrectly. Finding each requested term before adding avoids confusion between a term and a sum.

The governing concept is the explicit or general rule for a geometric progression. The formula is a_n = 2 × 5^(n−1), so substitute each required index carefully. For n = 4, a₄ = 2 × 5^3 = 2 × 125 = 250. For n = 5, a₅ = 2 × 5^4 = 2 × 625 = 1250. Adding the two terms gives a₄ + a₅ = 250 + 1250 = 1500. Therefore option A is correct. The exponent is n−1, not n; replacing it by n would make each calculated term five times too large. Options B, C, and D do not follow from the stated rule and can arise from using an incorrect exponent or adding incorrectly.

The governing concept is the nth-term formula for a geometric progression: a_n = a_1r^(n−1). Here a_1 = 8 and the common ratio is r = −16/8 = −2. For the seventh term, a_7 = 8(−2)^(7−1) = 8(−2)^6. The exponent is even, so (−2)^6 = 64, giving a_7 = 8 × 64 = 512. Hence option C is correct. The signs alternate because the ratio is negative: odd-position terms are positive and even-position terms are negative, so the seventh term must be positive. Options A and B have an incorrect magnitude, while D has the wrong sign despite having the correct magnitude pattern.

The governing relation is a_n = a_1r^(n−1). With r = 3, the second term is a_2 = a_1 × 3 = 3a_1. The fourth term is a_4 = a_1 × 3^3 = 27a_1 because three ratio-steps separate the first and fourth terms. Substituting into the given condition gives 3a_1 + 27a_1 = 90, so 30a_1 = 90 and a_1 = 3. Therefore option B is correct. A common mistake is to use 3^2 for the fourth term or to treat the sequence as arithmetic. Checking the result, the first terms are 3, 9, 27, 81 and 9 + 81 = 90, confirming the answer.