设数列{an}的前n项和为Sn=an-×2n+1+（n=1，2，3…），求首项a1和通项an．
付晨一生T勶
∵数列{an}的前n项和为Sn=an-×2n+1+（n=1，2，3…），∴当n=1时，a1=S1=1-13×22+23，解得a1=2．当n≥2时，an=Sn-Sn-1=an-×2n+1+-n-1-13×2n+23)，化为n＝4an-1+2n，化为n+2n=n-1+2n-1)，∴数列{n+2n}是等比数列，首项为4，公比q=4．∴n+2n＝4n，即n＝4n-2n．
为您推荐：
其他类似问题
当n=1时，a1=S1=1-13×22+23，解得a1=2．当n≥2时，an=Sn-Sn-1，转化为n＝4an-1+2n，化为n+2n=n-1+2n-1)，再利用等比数列的通项公式即可得出．
本题考点：
数列递推式．
考点点评：
本题考查了利用递推式求数列的通项公式，考查了转化为等比数列的数列的通项公式的求法，考查了推理能力与计算能力，属于难题．
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Enter your email to get latest Linux Howto's（2014o揭阳二模）已知等比数列{an}满足：a2=4，公比q=2，数列{bn}的前n项和为Sn，且Sn=bn-an+（n∈N*）．（1）求数列{an}和数列{bn}的通项an和bn；（2）设Pn=nSn（n∈N*），证明：Pi＜．
（1）由a2=4，q=2得，an＝a2o2n-2＝2n．（2分）由上式结合Sn＝43b&n-23an+23，得Sn＝43b&n-23(2n-1)，则当n≥2时，bn=Sn-Sn-1=43bn-23(2n-1)-43bn-1+23(2n-1-1)，（4分）∴bn-2n+1-4bn-1+2n=0...
为您推荐：
（1）由已知得n＝a2o2n-2＝2n．所以n＝43b&n-23(2n-1)，由此推导出数列n+2n}是首项为b1+2=4，公比为4的等比数列，从而求出n＝4n-2n．（2）由n＝4n-2n，得n＝anSn＝2n23(2n+1-1)(2n-1)＝32(12n-1-12n+1-1)，由此能证明Pi＜．
本题考点：
数列的求和；数列递推式；数列与不等式的综合．
考点点评：
本题考查数列的通项公式的求法，考查不等式的证明，解题时要认真审题，注意裂项求和法的合理运用．
扫描下载二维码给定数列，，，，，，，，，，按规律：（1）该数列的第20项是几？（2）是数列的第几项？（3）求前20项之和．
abTZ63MR19
（1）该数列的第20项是；（2）是数列的第33项；（3）++++++++++…+=（++…+）+（++…+）+（+++）+（++）+（+）=21++
为您推荐：
此题我们可以看出：分子分母相加为2，这组就有1个数，分子分母相加为3，这组就有2个数，…分子分母相加为n，这组就有n-1个数，分母从1逐渐增大1，分子逐渐减少1；（1）因为1+2+3+4+5+6=21，所以分子分母的和为7，第21项是，前面的一项是；（2）分子分母的和为9，是这一组中的第5个数，分子分母的和为8的前面一共有1+2+3+4+5+6+7=28，所以是数列的第28+5=33项；（3）写出前20项，按分母分类求和即可．
本题考点：
规律型：数字的变化类．
考点点评：
此题考查数字的变化规律，找出数字之间的联系，找出规律解决问题．
扫描下载二维码The following list provides an explanation of system errors -299 to -5553. This document applies to Mac OS 9 and earlier.
This article has been archived and is no longer updated by Apple.
1. Items listed in [brackets] are old result code meanings.
This information was not available.
Notification Manager Error
Invalid qType*must be ORD(nmType)
Start Manager Errors
smSDMInitErr
SDM could not be initialized.
smSRTInitErr
Slot Resource Table could not be initialized.
smPRAMInitErr
Slot Resource Table could not be initialized.
smPriInitErr
Cards could not be initialized.
smEmptySlot
No card in slot
CRC check failed for declaration data
smFormatErr
FHeader Format is not Apple's
smRevisionErr
Wrong revision level
Directory offset is Nil
smLWTstBad
Long Word test field && $5A932BC7.
smNosInfoArray
No sInfoArray. Memory Mgr error.
smResrvErr
Fatal reserved error. Reserved field && 0.
smUnExBusErr
Unexpected Bus Error
smBLFieldBad
ByteLanes field was bad.
smFHBlockRdErr
Error occurred during _sGetFHeader.
smFHBlkDispErr
Error occurred during _sDisposePtr (Dispose of FHeader block).
smDisposePErr
_DisposePointer error
smNoBoardsRsrc
No Board sResource.
smGetPRErr
Error occurred during _sGetPRAMRec (See SIMStatus).
smNoBoardId
No Board Id.
smIntStatVErr
The InitStatusV field was negative after primary or secondary init.
smIntTblVErr
An error occurred while trying to initialize the Slot Resource Table.
smNoJmpTbl
SDM jump table could not be created.
smBadBoardId
BoardId was wrong, re-init the PRAM record.
smBusErrTO
BusError time out.
smBadRefId
Reference Id not found in list
smBadsList
Bad sResource [sList] structure: Id1
smReservedErr
Reserved field not zero
smCodeRevErr
Code revision is wrong
Code revision is wrong
smsPointerNil
LPointer is nil {From sOffsetData. If this error occurs, check sInfo rec for more information.}
smNilsBlockErr
Nil sBlock error {Don't allocate and try to use a nil sBlock}
smSlotOOBErr
Slot out of bounds error
smSelOOBErr
Selector function not implemented
_NewPtr error
smBlkMoveErr
_BlockMove error
smCkStatusErr
Status of slot = fail.
smGetDrvrNamErr
Error occurred during _sGetDrvrName.
smDisDrvrNamErr
Error occurred during _sDisDrvrName.
smNoMoresRsrcs
No more sResources *OR*
smNoMoresResources
Specified sResource data structure not found
smsGetDrvrErr
Error occurred during _sGetDriver.
smBadsPtrErr
Bad pointer was passed to sCaldSPointer function
smByteLanesErr
ByteLanes field in card's format block was determined to be zero
smOffsetErr
Offset was too big (temporary, should be fixed)
smNoGoodOpens
No opens were successful in the loop.
smSRTOvrFlErr
SRT over flow.
smRecNotFnd
Record not found in the SRT.
Device Manager Slot Support Errors
slotNumErr
invalid slot # error
gcrOnMFMErr
gcr format on high density media error
Edition Manager Errors
editionMgrInitErr
Manager not initialized or could not load package
badSectionErr
Not a valid section type
notRegisteredSectionErr
Not registered
badSubPartErr
Bad edition container spec or invalid edition container
multiplePublisherWrn
Already is a publisher
containerNotFoundWrn
Alias was not resolved
notThePublisherWrn
Not the publisher
SCSI Manager Errors
scsiBadPBErr
invalid field(s) in the parameter block
scsiOverrunErr
attempted to transfer too many bytes
scsiTransferErr
write flag conflicts with data transfer phase
scsiBusTOErr
bus error during transfer
scsiSelectTOErr
scsiSelTO exceeded (selection failed)
scsiTimeOutErr
scsiReqTO exceeded
scsiBusResetErr
the bus was reset, so your request was aborted
scsiBadStatus
non-zero (not &Good&) status returned
scsiNoStatusErr
device did not go through a status phase
scsiLinkFailErr
linked command never executed
scsiUnimpVctErr
unimplemented routine was called
SysErrs used instead of inline $A9FF & $ABFF
user debugger break
strUserBreak
user debugger break - display string on stack
exUserBreak
user debugger break - execute commands on stack
QuickDraw Error
rgnTooBigErr
Bitmap would convert into a region greater than 64 KB
Text Edit Error
teScrapSizeErr
scrap item too big for text edit record
hwParamrErr
bad selector for _HWPriv
Processes Errors
procNotFound
No eligible process with specified process serial number
memFragErr
Not enough room to launch application with special requirements
appModeErr
Memory mode is 32-bit, but application is not 32-bit clean
protocolErr
app made module calls in improper order
hardwareConfigErr
hardware configuration not correct for call
appMemFullErr
Partition size specified in 'SIZE' resource is not big enough for launch
appIsDaemon
Application is background-only
bufferIsSmall
Buffer is too small
noOutstandingHLE
No outstanding high-level event
connectionInvalid
Connection is invalid
noUserInteractionAllowed
Attempted PostHighLevelEvent from background and no session yet established
Memory Dispatch Errors
notEnoughMemoryErr
Insufficient physical memory
notHeldErr
Specified range of memory is not held
cannotMakeContiguousErr
Cannot make specified range contiguous
notLockedErr
Specified range of memory is not locked
interruptsMaskedErr
Called with interrupts masked
DatabaseAccess (Pack 13) Errors
The data item was NULL
Data available or successfully retrieved
Error executing function
rcDBBadType
Next data item not of requested data type
Function timed out
Query currently executing
reDBBadSessID
Session ID is invalid
rcDBBadSessNum
Invalid session number
rcDBBadDDEV
Couldn't find the specified database extension, or error occurred in opening database extension
rcDBAsyncNotSupp
The database extension does not support asynchronous calls
rcDBBadAsyncPB
Invalid parameter block specified
rcDBNoHandler
There is no handler for this data type installed for the current application
rcDBWrongVersion
Wrong version number
Help Manager Errors
hmHelpDisabled
Help balloons are not enabled
hmResNotFound
hmMemFullErr
hmBalloonAborted
Because of constant cursor movement, the help balloon wasn't displayed
hmSameAsLastBalloon
Menu and item are same as previous menu and item
hmHelpManagerNotInited
Help menu not set up
hmBadSelector
hmSkippedBalloon
No balloon content to fill in
hmWrongVersion
Wrong version of Help Manager resource
hmUnknownHelpType
Help message record contained a bad type
hmCouldNotLoadPackage
hmOperationUnsupported
Bad method parameter
hmNoBalloonUp
No balloon showing
hmCloseViewActive
User using CloseView won't let you remove balloon
AppleTalk * PPC Toolbox Errors
notInitErr
PPC Toolbox has not been initialized yet
nameTypeErr
Invalid or inappropriate locationKindSelector in location name
I Unable to open port or bad port reference number
noGlobalsErr
System unable to allocate memory, critical error
localOnlyErr
Network activity is currently disabled
destPortErr
Port does not exist at destination
sessTableErr
PPC Toolbox is unable to create a session
noSessionErr
Invalid session reference number
Bad parameter or invalid state for this operation
portNameExistsErr
Another port is already open with this name
noUserNameErr
User name unknown on destination machine
userRejectErr
Destination rejected the session request
noResponseErr
Unable to contact application
portClosedErr
The port was closed
sessClosedErr
The session has closed
badPortNameErr
PPC port record is invalid
noDefaultUserErr
User has not specified owner name in Sharing Setup control panel
notLoggedInErr
Default user reference number does not yet exist
noUserRefErr
Unable to create a new user reference number
networkErr
An error has occurred in the network
noInformErr
PPCStart failed because target application did not have an inform pending
authFailErr
User's password is wrong
noUserRecErr
Invalid user reference number
badServiceMethodErr
Service method is other than ppcServiceRealTime
badLocNameErr
Location name is invalid
guestNotAllowedErr
Destination port requires authentication
AppleTalk - NBP Errors
nbpBuffOvr
Buffer overflow in LookupName
nbpNoConfirm
Name not confirmed on ConfirmName
nbpConfDiff
Name confirmed at different socket
nbpDuplicate
Duplicate name exists already
nbpNotFound
Name not found on remove
ASP Errors (XPP driver)
BadVersNum
Server cannot support this
BufTooSmall
Buffer too small
NoMoreSess
No more sessions on server
No servers at that address
Parameter error
ServerBusy
Server cannot open another session
SessClosed
Session closed
Command block too big
Too many clients (server error)
AppleTalk - ATP Errors
Request to contact router failed: retry count exceeded
tooManyReqs
Too many concurrent requests
tooManySkts
Too many concurrent responding-sockets
Bad ATP-responding socket
badBuffNum
Bad response buffer number specified
No release received
cbNotFound
Control B no pending asynchronous calls
noSendResp
AddResponse issued without SendResponse
noDataArea
No data area for request to MPP
Data Stream Protocol - DSP driver Errors
errOpenDenied
Open request denied by recipient
errDSPQueueSize
Send or receive queue is too small
errFwdReset
Read terminated by forward reset
errAttention
Attention message too long
errOpening
Attempt to open connection failed
Bad connection state for this operation
errAborted
Request aborted by dspRemove or dspClose function
HFS Errors
fidNotFound
File ID not found [No file thread exists]
File id already exists
notAFileErr
Specified file is a directory
diffVolErr
Files on different volumes
catChangedErr
Catalog has changed and CatPosition may be invalid
desktopDamagedErr
The desktop database has become corrupted * the Finder will fix this, but if your application is not running with the Finder, use DTReset or DTDelete
AppleEvent Errors
errAECoercionFail
Data could not be coerced to the requested descriptor type
errAEDescNotFound
Descriptor record was not found
errAECorruptData
Data in an Apple event could not be read
errAEWrongDataType
Wrong descriptor type
errAENotAEDesc
Not a valid descriptor record
errAEBadListItem
Operation involving a list item failed
errAENewerVersion
Need a newer version of the Apple Event Manager
errAENotAppleEvent
Event is not an Apple event
errAEEventNotHandled
Event wasn't handled by an Apple event handler
errAEReplyNotValid
AEResetTimer was passed an invalid reply
errAEUnknownSendMode
Invalid sending mode was passed
errAEWaitCanceled
User canceled out of wait loop for reply or receipt
errAETimeout
Apple event timed out
errAENoUserInteraction
No user interaction allowed
errAENotASpecialFunction
Wrong keyword for a special function
errAEParamMissed
Handler did not get all required parameters
errAEUnknownAddressType
Unknown Apple event address type
errAEHandlerNotFound
No handler found for an Apple event or a coercion
errAEReplyNotArrived
Reply has not yet arrived
AppleTalk - ATP Errors
buf2SmallErr
partial data returned
No MPP error
Check sum error
extractErr
Extraction error
Read queue error
ATP length error
ATP bad response error
Record not found
sktClosedErr
Socket closed error
Print Manager w/LaserWriter
No free Connect Control Blocks available
Bad connection reference number
Request already active
Write request too big
Connection just closed
Printer not found, or closed
File Manager Extensions Errors
accessDenied
Incorrect access for this file/folder
DenyConflict
Permission/Deny mode conflicts with the current mode in which this fork is already open
NoMoreLocks
Byte range locking failure from Server
RangeNotLocked
Attempt to unlock an already unlocked range
AFP Errors (XPP driver)
afpAccessDenied
AFP Access denied
afpAuthContinue
AFP Authorization continue
AFP Bad UAM
afpBadVersNum
AFP Bad version number
afpBitmapErr
AFP Bit map error
afpCantMove
AFP Can't move error
afpDenyConflict
AFP Deny conflict
afpDirNotEmpty
AFP Dir not empty
afpDiskFull
AFP Disk full
afpEofError
AFP End-of-File error
afpFileBusy
AFP File busy
AFP Flat volume
afpItemNotFound
AFP Information not found
afpLockErr
AFP Lock error
afpMiscErr
AFP Misc error
afpNoMoreLocks
AFP No more locks
afpNoServer
AFP No server
afpObjectExists
AFP Object already exists
afpObjectNotFound
AFP Object not found
afpParmErr
AFP Parm error
afpRangeNotLocked
AFP Range not locked
afpRangeOverlap
AFP Range overlap
afpSessClosed
AFP Session closed
afpUserNotAuth
AFP User not authorized
afpCallNotSupported
AFP Call not supported
afpObjectTypeErr
AFP Object type error
afpTooManyFilesOpen
AFP Too many files open
afpServerGoingDown
AFP Server going down
afpCantRename
AFP Can't rename
afpDirNotFound
AFP Directory not found
afpIconTypeError
Size of new icon and one it replaces don't match
afpVolLocked
Volume is Read-Only
afpObjectLocked
The specified command could not be completed because the object is locked.
afpContainsSharedErr
The specified share point contains a share point.
afpIDNotFound
The specified ID could not be found.
afpIDExists
The specified ID already exists.
afpDiffVolErr
Equivalent to the Mac OS error code.
afpCatalogChanged
The catalog has changed and CatPosition may be invalid. No matches were returned.
afpSameObjectErr
EEquivalent to the Mac OS error code.
afpBadIDErr
The specified ID is invalid.
afpPwdSameErr
The new password is the same as the old password.
afpPwdTooShortErr
The specified password is too short.
afpPwdExpiredErr
The password has expired.
afpInsideSharedErr
The specified directory is inside a share point.
afpInsideTrashErr
The specified directory is in the Trash.
afpPwdNeedsChangeErr
The password needs to be changed the first time the user logs on.
afpPwdPolicyErr
The specified password violates a UAM's policy.
afpAlreadyLoggedOn
The specified user is already logged on.
SysEnvirons Errors
envNotPresent
SysEnvirons trap not present--returned by glue.
envBadVers
Version non-positive
envVersTooBig
Version bigger than call can handle
Gestalt Errors
gestaltUnknownErr
Could not obtain the response
gestaltUndefSelectorErr
Undefined selector
gestaltDupSelectorErr
Selector already exists
gestaltLocationErr
Function not in system heap