<< Chapter < Page Chapter >> Page >

For the following exercises, write the first four terms of the sequence.

a n = n ! n 2

First four terms: 1 , 1 2 , 2 3 , 3 2

Got questions? Get instant answers now!

a n = 3 n ! 4 n !

Got questions? Get instant answers now!

a n = n ! n 2 n 1

First four terms: 1 , 2 , 6 5 , 24 11

Got questions? Get instant answers now!

a n = 100 n n ( n 1 ) !

Got questions? Get instant answers now!

Graphical

For the following exercises, graph the first five terms of the indicated sequence

a n = { 4 + n 2 n if  n  is even 3 + n if  n  is odd

Got questions? Get instant answers now!

a 1 = 2 ,   a n = ( a n 1 + 1 ) 2

Graph of a scattered plot with points at (1, 2), (2, 1), (3, 0), (4, 1), and (5, 0). The x-axis is labeled n and the y-axis is labeled a_n.
Got questions? Get instant answers now!

a n = 1 ,   a n = a n 1 + 8

Got questions? Get instant answers now!

a n = ( n + 1 ) ! ( n 1 ) !

Graph of a scattered plot with labeled points: (1, 2), (2, 6), (3, 12), (4, 20), and (5, 30). The x-axis is labeled n and the y-axis is labeled a_n.
Got questions? Get instant answers now!

For the following exercises, write an explicit formula for the sequence using the first five points shown on the graph.

For the following exercises, write a recursive formula for the sequence using the first five points shown on the graph.

Graph of a scattered plot with labeled points: (1, 6), (2, 7), (3, 9), (4, 13), and (5, 21). The x-axis is labeled n and the y-axis is labeled a_n.

a 1 = 6 ,   a n = 2 a n 1 5

Got questions? Get instant answers now!

Technology

Follow these steps to evaluate a sequence defined recursively using a graphing calculator:

  • On the home screen, key in the value for the initial term a 1 and press [ENTER] .
  • Enter the recursive formula by keying in all numerical values given in the formula, along with the key strokes [2ND] ANS for the previous term a n 1 . Press [ENTER] .
  • Continue pressing [ENTER] to calculate the values for each successive term.

For the following exercises, use the steps above to find the indicated term or terms for the sequence.

Find the first five terms of the sequence a 1 = 87 111 ,   a n = 4 3 a n 1 + 12 37 . Use the> Frac feature to give fractional results.

First five terms: 29 37 , 152 111 , 716 333 , 3188 999 , 13724 2997

Got questions? Get instant answers now!

Find the 15 th term of the sequence a 1 = 625 ,   a n = 0.8 a n 1 + 18.

Got questions? Get instant answers now!

Find the first five terms of the sequence a 1 = 2 ,   a n = 2 [ ( a n 1 ) 1 ] + 1.

First five terms: 2 , 3 , 5 , 17 , 65537

Got questions? Get instant answers now!

Find the first ten terms of the sequence a 1 = 8 ,   a n = ( a n 1 + 1 ) ! a n 1 ! .

Got questions? Get instant answers now!

Find the tenth term of the sequence a 1 = 2 ,   a n = n a n 1

a 10 = 7 , 257 , 600

Got questions? Get instant answers now!

Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI-84, do the following.

  • In the home screen, press [2ND] LIST .
  • Scroll over to OPS and choose “seq(” from the dropdown list. Press [ENTER] .
  • In the line headed “Expr:” type in the explicit formula, using the [ X,T , θ , n ] button for n
  • In the line headed “Variable:” type in the variable used on the previous step.
  • In the line headed “start:” key in the value of n that begins the sequence.
  • In the line headed “end:” key in the value of n that ends the sequence.
  • Press [ENTER] 3 times to return to the home screen. You will see the sequence syntax on the screen. Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms.

Using a TI-83, do the following.

  • In the home screen, press [2ND] LIST .
  • Scroll over to OPS and choose “seq(” from the dropdown list. Press [ENTER] .
  • Enter the items in the order “Expr” , “Variable” , “start” , “end” separated by commas. See the instructions above for the description of each item.
  • Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms.

For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary.

List the first five terms of the sequence a n = 28 9 n + 5 3 .

Got questions? Get instant answers now!

List the first six terms of the sequence a n = n 3 3.5 n 2 +   4.1 n 1.5 2.4 n .

First six terms: 0.042 , 0.146 , 0.875 , 2.385 , 4.708

Got questions? Get instant answers now!

List the first five terms of the sequence a n = 15 n ( 2 ) n 1 47

Got questions? Get instant answers now!

List the first four terms of the sequence a n = 5.7 n + 0.275 ( n 1 ) !

First four terms: 5.975 , 32.765 , 185.743 , 1057.25 , 6023.521

Got questions? Get instant answers now!

List the first six terms of the sequence a n = n ! n .

Got questions? Get instant answers now!

Extensions

Consider the sequence defined by a n = 6 8 n . Is a n = 421 a term in the sequence? Verify the result.

If a n = 421 is a term in the sequence, then solving the equation 421 = 6 8 n for n will yield a non-negative integer. However, if 421 = 6 8 n , then n = 51.875 so a n = 421 is not a term in the sequence.

Got questions? Get instant answers now!

What term in the sequence a n = n 2 + 4 n + 4 2 ( n + 2 ) has the value 41 ? Verify the result.

Got questions? Get instant answers now!

Find a recursive formula for the sequence 1 ,   0 ,   1 ,   1 ,   0 ,   1 ,   1 ,   0 ,   1 ,   1 ,   0 ,   1 ,   1 ,   ...   . ( Hint : find a pattern for a n based on the first two terms.)

a 1 = 1 , a 2 = 0 , a n = a n 1 a n 2

Got questions? Get instant answers now!

Calculate the first eight terms of the sequences a n = ( n + 2 ) ! ( n 1 ) ! and b n = n 3 + 3 n 2 + 2 n , and then make a conjecture about the relationship between these two sequences.

Got questions? Get instant answers now!

Prove the conjecture made in the preceding exercise.

( n + 2 ) ! ( n 1 ) ! = ( n + 2 ) · ( n + 1 ) · ( n ) · ( n 1 ) · ... · 3 · 2 · 1 ( n 1 ) · ... · 3 · 2 · 1 = n ( n + 1 ) ( n + 2 ) = n 3 + 3 n 2 + 2 n

Got questions? Get instant answers now!

Questions & Answers

preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply
what is the standard form if the focus is at (0,2) ?
Lorejean Reply
a²=4
Roy Reply
hil
Roy Reply
hi
Roy Reply
A bridge is to be built in the shape of a semi-elliptical arch and is to have a span of 120 feet. The height of the arch at a distance of 40 feet from the center is to be 8 feet. Find the height of the arch at its center
Abdulfatah Reply
Practice Key Terms 8

Get the best Precalculus course in your pocket!





Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask