Sequences

The first differences between the given terms are 4, 7, 12, and 19. Their successive differences are 3, 5, and 7, so the next one is 9. Hence, the next first difference is 19 + 9 = 28, and the next term is 48 + 28 = 76. If 74 were chosen, the next first difference would be 26, not 28. In exams, list the first differences first and then identify the pattern in their differences.

Substitute 4 for n: \(a_4=2^4+3^4\). Thus, \(a_4=16+81=97\), so option C is correct. A value such as 95 may result from evaluating a power or the addition incorrectly. Exam tip: evaluate each power separately before adding them.

(\frac{1}{10^2}=\frac{1}{100}) is not smaller, and (\frac{1}{11^2}<\frac{1}{100}). So the first term is the (11)th.

Solve \(\frac{n}{2n-1}<\frac{3}{5}\). For \(n\geq1\), \(2n-1\) is positive, so cross-multiplication gives \(5n<3(2n-1)\), or \(5n<6n-3\). Hence, \(n>3\). The smallest integer value of \(n\) is 4, so the first such term is \(a_4\), the fourth term. Note that \(a_3=\frac{3}{5}\), which is equal to, not less than, \(\frac{3}{5}\). Exam tip: For “less than”, use \(<\); an equal value is not included.

This is a geometric sequence in which each term is twice the preceding term. Its first five terms are \(8,16,32,64,128\). Their sum is \(8+16+32+64+128=248\), so the average is \(\frac{248}{5}=49.6\). Therefore, 49.6 is correct. The option 48 is incorrect because the total 248 must be divided by 5 to find the mean. Exam tip: Write all the required terms first, then divide their sum by the number of terms.

The formula gives each term directly, but the sign of \((-1)^n\) changes according to whether the index is even or odd. For an even index, \((-1)^n=1\); for an odd index, \((-1)^n=-1\). This sign must be handled carefully because the formula subtracts the power, rather than adding it.

For \(n=12\), \(a_{12}=6(12)-(-1)^{12}=72-1=71\). For \(n=11\), \(a_{11}=6(11)-(-1)^{11}=66-(-1)=67\). Therefore \(a_{12}-a_{11}=71-67=4\). The result is not 6 because the alternating power contributes a change as well. Hence option A is correct.

In an AP, the difference between consecutive terms is constant: \(a_{n+1}-a_n=d\). A constant ratio identifies a GP, not an AP. Exam tip: write two successive differences and compare them.

Given \(a_n=3n^2-2n+5\), \(a_{12}=3(12)^2-2(12)+5=432-24+5=413\) and \(a_{13}=3(13)^2-2(13)+5=507-26+5=486\). Therefore, \(a_{12}+a_{13}=413+486=899\). Option 886 is incorrect because it uses an incorrect value of \(a_{13}\). Exam tip: substitute each value of \(n\) separately and evaluate the square and multiplication first.

The rule is: next term = (previous term × 3) + 3. The last given term is 282, so the next term is \((282\times3)+3=846+3=849\). Option 846 is only three times 282 and misses the required addition of 3. Exam tip: apply the stated recurrence rule step by step to the last term.

Given \(a_n=2n^3-n\), \(a_6=2(6)^3-6=432-6=426\) and \(a_4=2(4)^3-4=128-4=124\). Therefore, \(a_6-a_4=426-124=302\). The value 304 may result from an error in subtraction or in evaluating a cube. In an exam, calculate each term separately before finding their difference.

A sequence is an ordered list, so changing the order changes the sequence; for example, 2, 5 and 5, 2 are different. Terms may repeat. Exam tip: do not confuse a sequence with a set.

The differences between consecutive terms are \(13-10=3\), \(19-13=6\), \(31-19=12\), and \(55-31=24\). Each difference is double the previous one, so the next difference is \(48\). Therefore, the next term is \(55+48=103\). Option 101 is incorrect because it would require a difference of \(46\), which is not double 24. Exam tip: For such sequences, first list the consecutive differences and check their pattern.

Since \(n^2-1=(n-1)(n+1)\), we get \(a_n=\frac{(n-1)(n+1)}{n+1}=n-1\). Hence, \(a_{15}=15-1=14\). Option 15 results from missing the subtraction of 1. Exam tip: factorise the numerator first and then cancel the common factor.

The consecutive differences are \(4-1=3\), \(10-4=6\), \(20-10=10\), and \(35-20=15\). These are the triangular numbers \(3,6,10,15\); the increases between them are \(3,4,5\), so the next increase is \(6\). Hence, the next difference is \(15+6=21\), and the next term is \(35+21=56\). Choosing \(54\) would give a difference of \(19\), which does not follow the pattern. Exam tip: For such sequences, first list consecutive differences and then check their pattern.

Given \(a_n=n^2+3^n\), \(a_4=4^2+3^4=16+81=97\) and \(a_5=5^2+3^5=25+243=268\). Hence, \(a_4+a_5=97+268=365\). The value 375 can result from an error while evaluating the power or adding the terms. Exam tip: Substitute the value of \(n\) first, then calculate the square and exponent separately.

(13^2+6\times13+5=252), so (13) is not correct. (12) also gives (221), so the target value is inconsistent.

The successive differences are 3, 6, 9, 12, and 15; each difference increases by 3. Therefore, the next difference is 18. Hence, the next term is \(47+18=65\). Choosing 62 would keep the difference at only 15, so it cannot be the next term. Exam tip: write the successive differences separately before extending a sequence.