(11C) n-th term of a Geometric Sequence - Printable Version +- HP Forums (https://www.hpmuseum.org/forum) +-- Forum: HP Software Libraries (/forum-10.html) +--- Forum: General Software Library (/forum-13.html) +--- Thread: (11C) n-th term of a Geometric Sequence (/thread-10770.html) (11C) n-th term of a Geometric Sequence - Gamo - 05-21-2018 09:31 AM Nth term of a Geometric Sequence giving Initial Value (a) Common Ratio (r) Term (n) Formula: a sub n = ar^n-1 for every integer n ≥ 1 More information about Geometric Progression: https://en.wikipedia.org/wiki/Geometric_progression Example: a=2 r=3.14 n=14 Procedure: Set User Mode 2 A > 2 // Input initial value 3.14 B > 3.14 // Input common ratio 14 C > 14 // Input term D > 5769197.69 // answer on the 14th term What is the sequence number on the 1st and 2nd term on the above example? (1st Term) 1 C > D > 2 (2nd Term) 2 C > D > 6.28 The sequence are 2, 6.28, 19.72, 61.92,....... Program: Code: ``` LBL A  // Initial Value STO 1 RTN LBL B  // Common Ratio STO 2 RTN LBL C  // n-term STO 3 RTN LBL D RCL 2 X>0?   // Test Common Ratio for Positive or Negative value  GTO 1 RCL 3 RCL 3 2 ÷ FRAC 2 x X=0?    // Test n-th term for Even or Odd number GTO 3 GSB 2    // Odd integer of n-th term RCL 1 x RTN LBL 1    // Common Ratio is Greater than Zero RCL 2 RCL 3 1 - Y^X RCL 1 x RTN LBL 2    // identical routine for common ratio value Less than Zero RCL 2 RCL 3 1 - X<>Y CHS X<>Y Y^X RTN LBL 3    // Even integer of n-th term GSB 2 CHS RCL 1 x RTN``` Remark: The common ratio of a geometric sequence may be negative, resulting in an alternating sequence, with numbers switching from positive to negative and back. For instance 1, −3, 9, −27, 81, −243, ... is a geometric sequence with common ratio −3. Input 1 A, -3 B, and your choice of term [C] then [D] for answer. ------------------------------------------------------------------------------ This update version can be use for the following: LBL A // To display a geometric progression LBL B // To find from the n-th term LBL C // To find the sum of the giving n-th term Procedure: For geometric sequence: (a) ENTER (r) > f [A] > continue [R/S] For n-th term: (a) ENTER (r) ENTER (n) > f [B] For the Sum of given n-th term: (a) ENTER (r) ENTER (n) > f [C] Example: a=1 r= -3 n= selected term Sequence: 1, -3, 9, -27, 81, -243,...... Geometric Sequence: 1 ENTER -3 f [A] > 1 > [R/S] > -3 > [R/S] > 9 4th term: 1 ENTER -3 ENTER 4 f [B] > -27 Sum to 4th term: 1 ENTER -3 ENTER 4 f [C] > -20 Program: Code: ``` LBL A  // Geometric Sequence  ENTER ENTER R^ LBL 1 R/S x GTO 1 LBL B    // n-th term Rv Rv STO 3 Rv Rv 1 - GSB 2 RCL 3 x RTN LBL C    // Sum to the n-th term Rv STO 4 Rv STO 3 Rv Rv GSB 2 1 - RCL 3 x RCL 4 1 - ÷ RTN LBL 2 STO 1 X<>Y STO 2 0 X<>Y X>Y GTO 3 CHS STO 2 RCL 1 2 ÷ FRAC 0 X=Y GTO 3 RCL 2 RCL 1 Y^X CHS RTN LBL 3 RCL 2 RCL 1 Y^X RTN``` Gamo RE: (11C) n-th term of a Geometric Sequence - Dieter - 05-21-2018 12:47 PM (05-21-2018 09:31 AM)Gamo Wrote:  Formula: a sub n = ar^n-1 for every integer n ≥ 1 an = a1 · rn–1 (05-21-2018 09:31 AM)Gamo Wrote:   Code: ```LBL D RCL 2 X>0?   // Test Common Ratio for Positive or Negative value  GTO 1 ... X=0?    // Test n-th term for Even or Odd number GTO 3 GSB 2    // Odd integer of n-th term RCL 1 x RTN LBL 1    // Common Ratio is Greater than Zero ... LBL 2    // identical routine for common ratio value Less than Zero ... LBL 3    // Even integer of n-th term GSB 2 CHS RCL 1 x RTN``` Gamo, all these tests and subroutines are not required. You do not have to check whether r is negative or not, and if it is, also check if n is odd or even. All HP calculators I know can calculate integer powers of negative bases directly. So (–3,14)5 directly yields –305,24... without any problem. This means the more than 40 steps routine at LBL D can be replaced with this: Code: ```LBL D RCL 2 RCL 3 1 - Y^X RCL 1 x RTN``` That's it. If you really want to check for odd or even n and whether r is negative or not, you could do it much shorter. Simply calculate the power with |r| and change the sign if r<0 and n is even. For instance this way: Code: ```LBL D CF 0 RCL 2 X>0? GTO 1  // keep flag 0 clear for r>0 RCL 3 2 / FRAC   // for r<0 check if n is odd or even X=0?   // if n is even SF 0   // set flag 0 LBL 1 RCL 2 ABS    // calculate |r|^(n-1) RCL 3 1 - Y^X RCL 1 x F? 0 CHS    // change sign if r<0 and n even CF 0 RTN``` But again: all this is not required. You know I'm a big fan of using the stack instead of registers, so here's how I would do it: Code: ```LBL A ENTER 1 - X<>Y ENTER R↓ X<>Y Y^X x RTN``` The ENTER in the second line usually is not required, but this way the program will also run on an HP25. ;-) Usage: a1 [ENTER] r [ENTER] n [A] => an [x] => an+1 [x] => an+2 ... Example: 1 [ENTER] –3 [ENTER] 1 [A]   1 [x] –3 [x]   9 [x] –27 [x]   81 [x] –243 ... Dieter RE: (11C) n-th term of a Geometric Sequence - Gamo - 05-21-2018 03:42 PM Dieter Thank You I like the shorter solution for even or odd routine. Your version of the step through a geometric progression is good but my manual version is easier. Steps: (r) ENTER ENTER ENTER (a) x (multiply) keep repeat this x as desire Example: 1, −3, 9, −27, 81, −243, ... -3 ENTER ENTER ENTER 1 X (repeat multiplication) Gamo RE: (11C) n-th term of a Geometric Sequence - Dieter - 05-21-2018 04:06 PM (05-21-2018 03:42 PM)Gamo Wrote:  I like the shorter solution for even or odd routine. Maybe it has not become clear yet: You do not need this odd/even thing at all! The 11 step program does it all. (05-21-2018 03:42 PM)Gamo Wrote:  Your version of the step through a geometric progression is good but my manual version is easier. Steps: (r) ENTER ENTER ENTER (a) x (multiply) keep repeat this x as desire Sure... if you start at the beginning of the sequence, i.e. n=1. With the program you can start at, say, n=20 and continue from there. Dieter