nth term

Substitute n=10 into the formula. The first part is \(\frac{10(10+1)}{2}=\frac{10\times11}{2}=55\). This is the sum of the first ten natural numbers, or the tenth triangular number. The formula then adds the fixed constant 3, so \(a_{10}=55+3=58\).

Therefore option A is correct. The constant addition must be included after evaluating the fraction; stopping at 55 would give option B, but that is only the triangular-number part and not the complete term. No further sequence pattern is needed because the explicit formula directly gives the value. The supplied answer and explanation correctly calculate both the variable part and the constant part.

Write the terms as products of consecutive integers: \(12=3\times4\), \(20=4\times5\), \(30=5\times6\), and \(42=6\times7\). Hence, the \(n\)th term is \((n+2)(n+3)\). Expanding gives \(n^2+5n+6\), so option D is correct. Option A gives the first term as 12, but its second term is 19, not 20. Exam tip: For such sequences, factor consecutive terms to identify the pattern in \(n\).

For a term to be positive, \(8n-17>0\). Thus, \(n>\frac{17}{8}\), and the smallest natural number satisfying this is \(3\). Checking confirms that \(a_2=-1\), which is not positive, whereas \(a_3=7\), which is positive. Therefore, the third term is the first positive term. Exam tip: after solving the inequality, choose the smallest natural-number value of \(n\) that satisfies it.

Given \(a_n=an^2+bn+1\), putting \(n=1\) gives \(a+b+1=6\), so \(a+b=5\). Putting \(n=2\) gives \(4a+2b+1=17\), so \(2a+b=8\). Solving these equations gives \(a=3\) and \(b=2\). Hence, \(a_5=3(5)^2+2(5)+1=75+10+1=86\). Option 84 would result from omitting the constant term \(+1\). Exam tip: substitute the given term numbers first to form linear equations for the unknown coefficients.

(a_8=180) and (a_3=25), so the difference is (155). In such questions, find both terms separately and subtract.