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If f(x) = x^2+2x-1 and g(x) =(2x-3), find each of the following functions (a) f o g (b) g o f (c) g o g o g. Checkcircle Expert Answer. Want to see the step-by-step answer? Check out a sample Q&A here. Want to see this answer and more? Step-by-step answers are written by subject experts who are available 24/7.

Can you prove that #1^2+2^2+3^2+..+n^2=1/6n(n+1)(2n+1)#?

2 Answers

Explanation:

#'using the method of 'color(blue)'proof by induction'#

#'this involves the following steps '#

#• ' prove true for some value, say n = 1'#

#• ' assume the result is true for n = k'#

#• ' prove true for n = k + 1'# Xilisoft iphone magic platinum 5 7 280.

#n=1toLHS=1^2=1#

#'and RHS ' =1/6(1+1)(2+1)=1#

#rArrcolor(red)'result is true for n = 1'#

#'assume result is true for n = k'#

#color(magenta)'assume ' 1^2+2^2+ .. +k^2=1/6k(k+1)(2k+1)#

#'prove true for n = k + 1'#

#1^2+2^2+.+k^2+(k+1)^2=1/6k(k+1)(2k+1)+(k+1)^2#

#=1/6(k+1)[k(2k+1)+6(k+1)]#

#=1/6(k+1)(2k^2+7k+6)#

#=1/6(k+1)(k+2)(2k+3)#

#=1/6n(n+1)(2n+1)to' with ' n=k+1#

#rArrcolor(red)'result is true for n = k + 1'#

#rArr1^2+2^2+3^2+..+n^2=1/6n(n+1)(2n+1)#

Explanation:

Let, #S_n=1^2+2^2+3^2+..+n^2, &, , f(n)=n^3, n in NNuu{0}.#

#:. f(n)-f(n-1)=n^3-(n-1)^3.#

#because, a^3-b^3=(a-b)(a^2+ab+b^2), f(n)-f(n-1),#

#={n-(n-1)}{n^2+n(n-1)+(n-1)^2},#

#=(1)(n^2+n^2-n+n^2-2n+1),#

# rArr f(n)-f(n-1)=n^3-(n-1)^3=3n^2-3n+1;(n in NNuu{0}.#

#n=1 rArr 1^3-0^3=3(1)^2-3(1)+1;#
#n=2 rArr 2^3-1^3=3(2)^2-3(2)+1;#
#n=3 rArr 3^3-2^3=3(2)^2-3(2)+1;#
#vdots vdots vdots vdots vdots vdots vdots vdots vdots vdots#
#n=n rArr n^3-(n-1)^3=3(n)^2-3(n)+1;#

#'Adding, 'n^3-0^3=3{1^2+2^2+3^2+..+n^2}-3{1+2+3+..+n}+n,#

# :. n^3=3S_n-3Sigman+n, or, #

# n^3=3S_n-3/2n(n+1)+n, i.e.,#

#2n^3=6S_n-3n(n+1)+2n=6S_n-3n^2-3n+2n,#

# :. 2n^3+3n^2+n=6S_n,#

# :. 6S_n=n(2n^2+3n+1)=n(n+1)(2n+1),#

# rArr S_n=n/6(n+1)(2n+1).#

Enjoy Maths.!

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Getif 2.x Network Tool

Hot Topics

Connecting to an SNMP deviceCompiling additional MIBS
Browsing and Graphing

What it is - Getif is a free multi-functional Windows GUI based Network Tool written by Philippe Simonet. It is amongst other things, an excellent SNMP tool that allows you to collect and graph information from SNMP devices. These devices include (but are by no means limited to) Windows 2000 (using the SNMP4NT or SNMP4W2K or SNMP-Informant extension agents, of course!), and other OS's as well as devices manufactured by most major network companies (i.e. Cisco, 3COM, Dlink, Nokia, etc., etc.).

Philippe has kindly given me permission to distribute this excellent tool.

GetIf is much more than an SNMP browser however, with the ability to graph OID values over time, display the device's interface information, routing and ARP tables, as well as do basic port scans, Traceroutes, NSLookups, and IP Scans.

There are now two versions of Getif available for download here at SNMP4tPC. Version 2.2 is arguably the most prevalent version, however version 2.3.1 is now also available for those who wish to try it. Some of the features of v 2.3.1 include sliding scrollbars (yippee!). Many of you have asked for this. Be advised however, that the familiar Set/Walk and Add to graph buttons have changed. I would suggest you keep version 2.2 around in addition to installing version 2.3.1. Together they make a great team!

Download Getif v2.2 now by clicking here!

Download Getif v2.3.1 now by clicking here!

Since you're using Getif.. you might want to download THIS file! Gently unzip it into the MIBS directory below wherever you installed Getif (usually x:Program FilesGetif 2.xMIBS). Be SURE to BACK UP YOUR MIBS DIRECTORY FIRST! VERY IMPORTANT!

Once unzipped, you will see a PILE of MIBS (about 620) in the GetifMibs directory. You will also see a nice directory structure, where all the MIBS exist as well. This way, you can delete the MIB files you don't want from the GetifMIBS, and recover them later if you want from the individual directories.

If present, delete the .index file in GetifMIBS, and restart Getif, Click the MBrowser tab, and you ought to be quite impressed with what you find! This 8.0MB zip contains many MIBS that are known to work with Getif including BEA Weblogic, Cisco (comprehensive!), DOCSys compliant cable modems, AppleTalk, Adaptec, Madge, Stratacom, SymBIOS, Mylex, LanOptics, LightStream, DEC, NEC, IBM, FastTrack, GrandJunction, Microsoft, NetWare, GroupWise, APCC, Compaq Insight Manager, HP TopTools, and RFC MIBS. It also contains a number of MIB-II/Transmission types such as RS-232, 802.3, 802.5, FDDI, LAPB, X.25, Frame Relay, ISDN, DS1, DS3 and SoNET. It does NOT include the SNMP4tPC PERFMIB.MIB.

I have spent quite a while collecting these MIBS for Getif. They ought to work for most SNMPv1 compliant browsers as well.

Download the SNMP4tPC Getif MIB collection now by clicking here!

You will notice that the Getif.log file has some entries now. This is due in part to missing cross-references. I have not spent a lot of time cleaning it up, as it might reduce the number of browse-able objects. Plus, it's a lot of work. Remember, we're compiling 620 MIBS now!

Important Note: In order to use Getif effectively, you must have SNMP installed, and know the IP address and the community name(s) of the device you want to access. If you don't, the SNMP functions will not work.

The remainder of this page attempts to help you discover how to get the most out of Getif, and describes each tab in more detail. Skip it if all you want is the program and/or the MIB collection.

Parameters -To begin with, you must first start the program, and enter in the IP address of the device you want to connect to as well as the read and write community strings. You do this from the Parameters page as shown below.

On this page, you can also enter the SNMP timeout and retry values. You should leave them at their default values in most cases, however on a slow LAN or WAN, you might want to increase the timeout (I suggest increments of 500ms if you need to.) and retry values.

Note: The fields with white backgrounds can be changed; the rest are display fields only.

The ifNumber field identifies the number of interfaces found by Getif on the device selected. For example, on my system, I have 3 (1 loopback, and 2 real NICs).

The SysServices number identifies the SNMP support provided by the system. On my system, the number 76 represents the fact the the box is providing application, end-to-end, and Internet services. Each service type has a number assigned, and they add up to the SysServices number displayed. Sorry! I can't remember what each service number is (as I write this).

Hint: You can set the default IP Address, and community names here, so you don't have to re-enter them each time you connect

Interfaces - Selecting this tab and pressing Start will display a list of all the discovered interfaces on the selected device.

You can select to display All (default) or Administrative Up or Operationally Up interfaces only by selecting the appropriate check box.

Hint: Expand the int. field column (drag it wider) to make sure you get the entire interface number! On Windows 2000, this can be quite large (i.e. 167772##). This is the number you would plug into the MRTG Target[] line. For example, to monitor the internal network card on the system above, the MRTG target line would be:
Target[WIN2KFS1_LOCAL_NIC]: 16777220:cilbup@192.168.200.10

Addresses - Displays the interface address information. Press the Start button.

Routing Table - Displays the routing table for each interface. Press the Start button.

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Arp - Displays contents of the Arp Cache table. Press the Start button.

Gen. Table - displays the ipNetToMediaTable. Press the Start button.

IP Discovery - Here, you can select an IP Address range, and scan it to see what hosts are active. You can control the initial timeout, and subsequent timeouts, etc. so you don't spend all day scanning for non-existent devices. You can also select whether or not to resolve names. When ready, press the Start button.

Reachability - Select this tab, and press Start to scan the selected device for the services displayed on the left hand side of the pane. This is a handy way to quickly check for common services on the target device. When ready, press the Start button.

Hint: You can also choose to do a Ping scan as well, by clicking the check box. If you want to add the Ping scan time plots on a graph, press the Add to graph button. Then, go to the Graph tab, and press Start. You will see your PING scan times plotted there. The PING will continue as long as the check box is checked.

MIB Browser -Getif will compile SNMPv1 MIBS (it will not work with SNMPv2/3), and let you browse them in a graphical tree format as can be seen in the image below, which shows Getif browsing the Memory counters (from SNMP4W2K) on my Windows 2000 Advanced Server system. It shows the currently available (free) memory in bytes.

The top field displays the fully qualified SNMP OID name. The second field displays the representative SNMP OID number (as determined by the MIB compile). The top main screen shows the tree-view of the MIB. By selecting the value you want, and then pressing the Walk (also used as GET) button, the lower main screen will fill with the result of the walk. Select the instance you want, and the bottom 3 fields will be populated. The bottom left field displays the numeric OID (including the instance number). The middle window diplays the SNMP variable type (ipaddress, integer, nullobj, objid, string, timeticks or unsigned, and finally, the bottom right field displays the retrieved value.

Hint: If you press the Add to Graph button, the selected instance will be copied to the Graph Tab for you to do real-time monitoring. You can do this with more than one instance. Just be sure the scale gradient (i.e. %, bytes, etc.) is functional for whatever OIDs you select.

There are some limitations in version 2.2 of Getif, one being that you cannot scroll to the right in the lower main window.

Compiling additional MIBS - In order to add new (SNMPv1) MIBS to the Getif browser, you must perform the following steps.

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1. Be sure to shut down Getif
2. Copy your new MIB(s) into the MIBS directory under the Getif install directory (usually C:Program FilesGetif 2.2MIBS). Be sure that any requisite MIBS (some MIBS require that other MIBS are there) are also copied!
3. delete the .index file in the the C:Program FilesGetif 2.2MIBS directory
4. Restart Getif

Check the C:Program FilesGetif 2.2Getif.log file for any errors. If no errors are found during the compile, the file will be 0 (zero) bytes in size.

Hint: I have created a MIBS.ZIP (available here) file containing a TON of additional files! Check it out!

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Graph - Basically, once you have added an SNMP (or maybe a PING) value to the graph, you must go to the Graph page and press Start. The graph will then start plotting based on the Refresh time you select on the page. You can choose to have the labels displayed, or have the graph show in 3D mode (not my favorite). In this particular example, I am plotting the available memory (in bytes) as added to the graph from the Mib Browser Tab (above)

Hint: Choosing the X Labels will show the real (not relative) time of each plot. the Nb pts increases or decreases the granularity.

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