Sequences

Use the recurrence step by step. Thus, a_2=2(2)+1^2=5, a_3=2(5)+2^2=14, and a_4=2(14)+3^2=37. Therefore, 37 is correct. Although 39 is a close distractor, it is not the correct value of 2(14)+3^2. Exam tip: to find a_4, substitute n=1, then n=2, and then n=3; do not substitute n=4 directly.

In option B, the sequence falls from 1 to -1 and then rises from -1 to 1, so it does not move in one direction. A and D increase, while C decreases. Exam tip: compare consecutive terms.

(T_{19}=190) and (T_{20}=210), so the first greater term is the (20)th. Exam tip: strict inequality does not include equality.

The differences between consecutive terms are \(21-9=12\), \(39-21=18\), \(63-39=24\), and \(93-63=30\). These differences increase by \(6\) each time, so the next difference is \(36\). Therefore, the next term is \(93+36=129\). Option 123 would result if the difference remained 30, but the pattern shows that it continues increasing. Exam tip: For such sequences, list the first and second differences to identify the pattern.

The terms are (5,13,32,73,158), so (a_5=158). Therefore none of the given options is correct.

The differences between consecutive terms are \(13-4=9\), \(28-13=15\), \(49-28=21\), and \(76-49=27\). These differences increase by \(6\) each time, so the next difference is \(27+6=33\). Therefore, the next term is \(76+33=109\). Option \(107\) would require the next difference to be \(31\), which does not follow the given difference pattern. Exam tip: For such sequences, first list consecutive differences and then look for their pattern.

Its terms are \(3,5,9,17,\ldots\). The consecutive differences \(2,4,8\) are not equal, and the ratios \(5/3,9/5\) are also not equal. Hence it is neither an AP nor a GP. Exam tip: check differences for AP and ratios for GP.

(2\times13^2+5\times13=403), so it is the (13)th term. Exam tip: substitute the options into the formula.

The consecutive differences are \(146-150=-4\), \(137-146=-9\), \(123-137=-14\), and \(104-123=-19\). These differences decrease by \(-5\) each time, so the next difference is \(-24\). Therefore, the next term is \(104-24=80\). Choosing 82 would give a difference of \(-22\), which does not follow the difference pattern. Exam tip: For such sequences, first list the consecutive differences and check their pattern.

Direct answer: Option D, 98. The operations alternate: first add 3, then multiply by 2, and repeat. Starting with 7, add 3 to get 10. Multiply 10 by 2 to get 20. Add 3 to get 23. Multiply by 2 to get 46. Add 3 to get 49. The next operation after this addition is multiplication by 2, so 49×2=98. Option A, 52, would result from incorrectly adding 3 again. Option B, 92, could come from doubling 46 again or from using the wrong previous term, so it does not follow the stated pattern. Option C, 95, is not produced by either required operation from 49. Option D, 98, correctly applies the next multiplication. Memory cue: mark each step as +3, ×2, +3, ×2; after 49, the next symbol is ×2.

In the recurrence, add the square of the current value of n at every step. Thus, a_2=2(3)+1^2=7, a_3=2(7)+2^2=18, a_4=2(18)+3^2=45, and a_5=2(45)+4^2=106. Therefore, 106 is correct. A value such as 107 can result from an error while adding 4^2. Exam tip: to find a_5, apply the rule successively for n=1, 2, 3, and 4.

The general term is (\frac{2n+1}{3n+1}), so the (9)th term is (\frac{19}{28}). Exam tip: observe numerator and denominator patterns separately.

Given \(a_n=3^n-n^2\), substitute \(n=5\): \(a_5=3^5-5^2=243-25=218\). Hence, 218 is the correct answer. The nearby option 214 is incorrect because \(3^5=243\), not 239. Exam tip: substitute the term number first, then evaluate the power and the square separately.

(a_8=576) and (a_6=252), so the difference is (324), not (336). Recheck cube and square calculations carefully.

Given \(a_n=n^3+n^2\), \(a_8=8^3+8^2=512+64=576\) and \(a_6=6^3+6^2=216+36=252\). Hence, \(a_8-a_6=576-252=324\). The value 312 can result from an error while calculating a cube or a square. Exam tip: evaluate each required term separately before subtracting.

The successive differences 4, 9, 16, and 25 are \(2^2,3^2,4^2,5^2\), respectively. Hence, the next difference is \(6^2=36\). Therefore, the next term is \(59+36=95\). Getting 90 would require adding 31, which does not follow the square-difference pattern. Exam tip: check successive differences to spot patterns involving consecutive squares quickly.

The direct answer is option B: 9 terms. We need positive-indexed terms satisfying \\(a_n=5n^2-4n+1<400\\). First check the boundary values. For n=9, \\(a_9=5(9^2)-4(9)+1=5×81-36+1=405-36+1=370\\), which is less than 400. For n=10, \\(a_{10}=5(10^2)-4(10)+1=500-40+1=461\\), which is greater than 400. Since the quadratic expression increases for the positive integers here, all terms from n=1 through n=9 are less than 400, and the tenth and later terms are not. Thus there are 9 terms. Option A, 8, stops too early because the ninth term is still below 400. Option B, 9, is correct. Option C, 10, is wrong because the tenth term is 461, not less than 400. Option D, 11, is even farther beyond the boundary. Direct checks at the boundary are safest. Do not confuse “less than 400” with “less than or equal to 400”; even with equality allowed, the count would not change here, but the wording still requires a strict comparison.

The difference between the second and third terms is 12, so x = 11 + 12 = 23. Check: 41 - 23 = 18 and 66 - 41 = 25, so all the given successive differences match. If 25 were chosen, the difference up to 41 would be 16, not 18. Exam tip: after finding a missing term, verify the differences on both sides.

The third term is 221 and the next difference is -25. Therefore, x = 221 + (-25) = 196. Checking further, 196 + (-30) = 166, so the following given difference also fits. Option 190 is incorrect because the difference from 221 to 190 is -31, not -25. Exam tip: adding a negative difference means the term decreases.

Substituting \(n=18\), \(a_{18}=\frac{18^2+18}{2}=\frac{324+18}{2}=\frac{342}{2}=171\). Hence, 171 is the correct option. \(190\) is the next term, \(a_{19}\), since \(\frac{19\times20}{2}=190\). Exam tip: substitute the given value of \(n\) first, then calculate the square and addition carefully.

Given \(a_n=2n!+1\), we get \(a_5=2\times5!+1\). Since \(5!=5\times4\times3\times2\times1=120\), \(a_5=2\times120+1=241\). The value 121 would result from using \(5!+1\) and incorrectly omitting the multiplication by 2. Exam tip: evaluate the factorial before carrying out the remaining operations.

Substituting \(n=7\), \(a_7=\frac{3(7)-1}{2(7)+5}=\frac{21-1}{14+5}=\frac{20}{19}\). Therefore, option A is correct. \(\frac{19}{20}\) results from reversing the numerator and denominator. Exam tip: substitute the term number first, then simplify the numerator and denominator separately.

In a sequence with equal differences, the differences between consecutive terms are equal. Therefore,
\((2x+3)-(x-1)=(4x+1)-(2x+3)\).
The left side simplifies to \(x+4\), while the right side simplifies to \(2x-2\). Hence, \(x+4=2x-2\), giving \(x=6\). If 5 is used, the two differences are not equal. Exam tip: For three consecutive terms, set second term − first term equal to third term − second term.

For sequence B, the first differences are 3, 5 and 7, so they are not equal. The second differences are \(5-3=2\) and \(7-5=2\). Thus B is correct. In exams, list two rows of differences to identify such a sequence.

For \(a_n=5n-2\), \(a_{n+1}-a_n=[5(n+1)-2]-(5n-2)=5\), which is constant. Hence it is an arithmetic progression. In \(n^2+1\), the difference changes. Exam tip: check consecutive differences.