For problems #1-9 write the explicit formula of an arithmetic sequence from the given information
1) The first term is \(11\) and the common difference is \(4\)
2) \(a_1 = 5\) and \(d=-1\)
3) \(a_3=4\) and \(d=6\)
4) \(a_1=9\) and \(a_{12}= 86\)
5) \(a_5=10\) and \(a_{13}=-38\)
6) \(a_2= 7\) and \(a_3=10\)
7) \(d=6\) and \(a_7=1\)
8) \(a_4=4\) and \(a_{10}=10\)
9) \(a_5=5\) and \(a_7=5\)
For problems #10-12 determine if the sequence is arithmetic or not. If so, state the common difference, if not, explain why.
10) \(8, 4, 0, -4, -8, ...\)
11) \(1, 3, 9, 27, 81, ...\)
12) \(\frac{7}{3},\frac{5}{3},1,\frac{1}{3},...\)
13) One term of a sequence is \(4\), say \(a_i=4\). If you double the term number you get \(9\), so \(a_{2i}=9\). Find the arithmetic sequence with the largest common difference satisfying these conditions.
14) In arithmetic sequence the first term is \(14\) and the seventh term is \(62\). Find the \(42\)nd term.
15) In an arithmetic sequence if \(a_2=-5\) and \(a_{53}=158\), find \(a_{107}\).
16) Find the sum of the first \(15\) terms of the arithmetic sequence given by \(a_n=4+7(n-1)\).
17) Evaluate the sum \(\sum\limits_{k=1}^{155} -3k+9\), explain why the sum is a negative number.
18) You are given a summer tutoring job. You agree to get paid \(\$200\) per week to begin with the understanding that you will get a \(\$10\) raise every week. If you work for \(10\) weeks this summer how much money will you make altogether?
19) In an auditorium there are \(15\) seats in the first row, one additional seat is added to each end of every next row. If there are \(22\) rows of seats in the auditorium, how many total seats are there?
20) Find the sum of the first \(500\) consecutive positive integers.
Review
21) If the half life of carbon-14 is \(5730\) years and it is known that a substance began with \(200\) grams of carbon-14, determine how much carbon-14 will be present in the substance if it has aged \(15000\) years. Round to the nearest hundredth of a gram.
22) Find the inverse of \(f(x)=5 \ln (x-3)\).
23) If \(f\) is an injective (one-to-one) function with Dom \(f=\{x:x \geq 0 \}\) and Rng \(f=\{y: y<3\}\) state the domain and range of \(f^{-1}\) and state how you know \(f^{-1}\) exists.
24) Condense into a single logarithm: \(3 \ln 2- \ln y + \frac{1}{2} \ln z\).
25) Which is greater, \(\log_{2} 9\) or \(\log_{3} 16\)? State how you know, do not use a calculator.
Solution Bank