11.1E: Exercises for Section 11.1

In exercises 1 - 4, find the first six terms of each sequence, starting with \( n=1\).

1) \(a_n=1+(−1)^n\) for \( n≥1\)

Answer

\( a_n=0\) if \( n\) is odd and \( a_n=2\) if \( n\) is even

2) \( a_n=n^2−1\) for \( n≥1\)

Answer

\( {a_n}={0,3,8,15,24,35,…}\)

3) \( a_1=1\) and \( a_n=a_{n−1}+n\) for \( n≥2\)

Answer

\( {a_n}={1,3,6,10,15,21,…}\)

4) \( a_1=1, a_2=1\) and \( a_{n+2}=a_n+a_{n+1}\) for \( n≥1\)

Answer

\( {a_n}={1,1,2,3,5,8, …}\)

5) Find an explicit formula for \( a_n\) where \( a_1=1\) and \( a_n=a_{n−1}+n\) for \( n≥2\).

Answer

\( a_n=\dfrac{n(n+1)}{2}\)

6) Find a formula \( a_n\) for the \(n^{\text{th}}\) term of the arithmetic sequence whose first term is \( a_1=1\) such that \( a_{n+1}−a_n=17\) for \( n≥1\).

Answer

\( a_n=17n-16 \)

7) Find a formula \( a_n\) for the \(n^{\text{th}}\) term of the arithmetic sequence whose first term is \( a_1=−3\) such that \( a_{n+1}−a_n=4\) for \( n≥1\).

Answer

\( a_n=4n−7\)

8) Find a formula \( a_n\) for the \(n^{\text{th}}\) term of the geometric sequence whose first term is \( a_1=1\) such that \( \dfrac{a_{n+1}}{a_n}=10\) for \( n≥1\).

Answer

\( a_n=10^{n-1} \)

9) Find an explicit formula for the \(n^{\text{th}}\) term of the sequence whose first several terms are \( {0,3,8,15,24,35,48,63,80,99,…}.\) (Hint: First add one to each term.)

Answer

\( a_n=n^2-1\)

10) Find an explicit formula for the \(n^{\text{th}}\) term of the sequence satisfying \( a_1=0\) and \( a_n=2a_{n−1}+1\) for \( n≥2\).

Answer

\( a_n=2^{n-1}−1\)

In exercises 11 and 12, find a formula for the general term \( a_n\) of each of the following sequences.

11) \( {1,0,−1,0,1,0,−1,0,…}\) (Hint: Find where \(\sin x\) takes these values)

Answer

\( a_n=\sin \left( \frac{n\pi}{2} \right) \)

12) \( {1,−1/3,1/5,−1/7,…}\)

Answer

\( a_n=\dfrac{(−1)^{n−1}}{2n−1}\)

Find a function \( f(n)\) that identifies the \(n^{\text{th}}\) term \( a_n\) of the following recursively defined sequences, as \( a_n=f(n)\).

13) \( a_1=2\) and \( a_{n+1}=2a_n\) for \( n≥1\)

Answer

\( f(n)=2^n\)

In exercises 14-16, find the limit of each of the following sequences, using L’Hôpital’s rule when appropriate.

14) \( \dfrac{n^2}{2^n}\)

Answer

\(\displaystyle \lim_{n→∞} \dfrac{n^2}{2^n} \quad = \quad 0\)

15) \( \dfrac{\sqrt{n}}{\sqrt{n+1}}\)

Answer

\(\displaystyle \lim_{n→∞} \dfrac{\sqrt{n}}{\sqrt{n+1}} \quad = \quad 1 \)

16) \( n^{1/n}\) (Hint: \( n^{1/n}=e^{\frac{1}{n}\ln n})\)

In exercises 17-21, state whether each sequence is bounded and whether it is eventually monotone, increasing, or decreasing.

17) \( n/2^n, n≥2\)

Answer

bounded, decreasing for \( n≥1\)

18) \( \ln\left(1+\dfrac{1}{n}\right)\)

Answer

bounded, decreasing

19) \( \sin n\)

Answer

bounded, not monotone

20) \( n^{1/n}, \quad n≥3\)

Answer

bounded, decreasing

21) \( \tan n\)

Answer

not monotone, not bounded

In exercises 22-23, determine whether the given sequence has a limit. If it does, find the limit. Use the Squeeze Theorem to find the limit of each sequence in exercises 24 - 25.

22) \( a_1=\sqrt{2}, a_2=\sqrt{2\sqrt{2}}. a_3=\sqrt{2\sqrt{2\sqrt{2}}}\) etc.

Answer

2

23) \( a_1=3, a_n=\sqrt{2a_{n−1}}, n=2,3,….\)

Answer

\( a_n\) is decreasing and bounded below by \(2\). The limit a must satisfy \( a=\sqrt{2a}\) so \( a=2\), independent of the initial value.

24) \( n\sin(1/n)\)

Answer

Limit is 1.

25) \( \dfrac{\cos(1/n)−1}{1/n}\)

Answer

\(0\)

In exercises 26 - 32, determine the limit of the sequence or show that the sequence diverges. If it converges, find its limit.

26) \( a_n=\tan^{−1}(n^2)\)

Answer

\( \pi/2 \)

27) \( a_n=(2n)^{1/n}−n^{1/n}\)

Answer

\( n^{1/n}→1\) and \( 2^{1/n}→1,\) so \( a_n→0\)

28) \( a_n=\dfrac{\ln(n^2)}{\ln(2n)}\)

Answer

Limit is 2.

29) \( a_n=\left(1−\frac{2}{n}\right)^n\)

Answer

Since \( (1+1/n)^n→e\), one has \( (1−2/n)^n≈(1+k)^{−2k}→e^{−2}\) as \( k→∞.\)

30) \( a_n=\ln\left(\dfrac{n+2}{n^2−3}\right)\)

Answer

Limit is 1.

31) \( a_n=\dfrac{2^n+3^n}{4^n}\)

Answer

\( 2^n+3^n≤2⋅3^n\) and \( 3^n/4^n→0\) as \( n→∞\), so \( a_n→0\) as \( n→∞.\)

32) \( a_n=\dfrac{(n!)^2}{(2n)!}\)

Answer

\( \dfrac{a_{n+1}}{a_n}=n!/(n+1)(n+2)⋯(2n) =\dfrac{1⋅2⋅3⋯n}{(n+1)(n+2)⋯(2n)}<1/2^n\). In particular, \( a_{n+1}/a_n≤1/2\), so \( a_n→0\) as \( n→∞\).

33) [T] Suppose you start with one liter of vinegar and repeatedly remove \(0.1\) L, replace with water, mix, and repeat.

a. Find a formula for the concentration after \( n\) steps.

b. After how many steps does the mixture contain less than \( 10\%\) vinegar?

34) [T] A lake initially contains \( 2000\) fish. Suppose that in the absence of predators or other causes of removal, the fish population increases by \( 6\%\) each month. However, factoring in all causes, \( 150\) fish are lost each month.

a. Explain why the fish population after \( n\) months is modeled by \( P_n=1.06P_{n−1}−150\) with \( P_0=2000\).

b. How many fish will be in the pond after one year?

Answer

a. Without losses, the population would obey \( P_n=1.06P_{n−1}\). The subtraction of \( 150\) accounts for fish losses.
b. After \( 12\) months, we have \( P_{12}≈1494.\)

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