Sequences

(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.

After (49), multiplication by (2) is applied, so (49\times2=98). Exam tip: identify the next operation in alternating-operation sequences.

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.

(a_9=370) and (a_{10}=461), so (9) terms are less than (400). Exam tip: always check terms near the boundary.

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.