Introduction to sequences
2000010310
Level:
A
The \(n\)th term of a sequence is \(\left(\frac12\right)^n\). Find the difference between the \(5\)th and the \(8\)th term of the sequence.
\( \frac{7}{256} \)
\( \frac{3}{128} \)
\(-\frac{7}{256} \)
\( 0\)
2000010309
Level:
A
A sequence has \(n\)th term \(50-\frac{1}{2}n^2\). Which term of the sequence is the first to have the value less than \(0\)?
\( a_{11} \)
\( a_{10} \)
\( a_{6} \)
\( a_{5}\)
2000010308
Level:
A
A sequence has \(n\)th term \(\frac{1}{3+2n}\) . Which term of the sequence is the first to have the value less than \(\frac1{200}\)?
\( a_{99} \)
\( a_{98} \)
\( a_{101} \)
\( a_{102}\)
2000010307
Level:
B
Which of the following sequences given by the recursive formulas is not decreasing?
\( a_{n+1} = \frac{1}{a_n}\), \( a_1=5\)
\( a_{n+1} = \sqrt{a_n}\), \( a_1=16\)
\( a_{n+1} = 0.5\cdot {a_n}\), \( a_1=12\)
\( a_{n+1} = \frac{a_n}{n}\), \( a_1=24\)
2010010306
Level:
B
Which of following statements about the sequence \( \left( \frac{n-2}{n+1}\right)^{\infty}_{n=1} \) is true?
\[\]
(Help: A sequence is bounded below if all its terms are greater than or equal to a real number \(L\), which is called the lower bound of the sequence. Similarly, a sequence is bounded above if all its terms are less than or equal to a real number \(U\), which is called the upper bound of the sequence.)
one of lower bounds is \(-\frac12\), one of upper bounds is \(1\)
one of lower bounds is \(-\frac12\), upper bound does not exist
lower bound does not exist, one of upper bounds is \(1\)
there exists neither lower bound nor upper bound
2010010305
Level:
B
Which of following statements about the sequence \( \left( \frac{n+3}{2n}\right)^{\infty}_{n=1} \) is true?
\[\]
(Help: A sequence is bounded below if all its terms are greater than or equal to a real number \(L\), which is called the lower bound of the sequence. Similarly, a sequence is bounded above if all its terms are less than or equal to a real number \(U\), which is called the upper bound of the sequence.)
one of lower bounds is \(\frac12\), one of upper bounds is \(2\)
one of lower bounds is \(\frac12\), upper bound does not exist
lower bound does not exist, one of upper bounds is \(2\)
there exists neither lower bound nor upper bound
2010010304
Level:
B
We are given a sequence \( \left( \frac{n+5}{n+1}\right)^{\infty}_{n=1}\). What are the properties of this sequence?
decreasing and bounded above
increasing and not bounded below
neither increasing nor decreasing
increasing and bounded below
decreasing and not bounded above
2010010303
Level:
B
We are given a sequence \( \left( \frac{2n+1}{n+3}\right)^{\infty}_{n=1}\). What are the properties of this sequence?
increasing and bounded
neither increasing nor decreasing
decreasing and bounded above
increasing and not bounded above
decreasing and not bounded above
2010010302
Level:
B
Which of the following sequences given by the formula for the \(n\)th term is not increasing?
\( (n^{-4})^{\infty}_{n=1}\)
\( (\sqrt{n})^{\infty}_{n=1}\)
\( \left( -\frac{n+1}{n}\right)^{\infty}_{n=1}\)
\( \left( {2^n}\right)^{\infty}_{n=1}\)