ACJC 2025 Sequences and Series Summary

Anglo-Chinese Junior College 2025 H2 Mathematics 9758: Sequences & Series / Summary / Page 1 of 3 SUMMARY: Sequences and Series Notations a: first term of a sequence, particularly in an arithmetic (AP) or geometric progression (GP). nu or nT : nth term of a sequence nS : sum of the first n terms of a sequence Special sequences: Arithmetic and Geometric Sequences AP GP Type a, a + d, a + 2d, … , a + ( 1)n− d, … a, ar, ar2, … , 1nar − , … Characteristic Common difference, d, i.e., 1 constantnnd u u −= − = Common ratio, r, i.e. 1 constantn n ur u − == nth term ( 1)nu a n d= + − 1n nu ar −= Sum of the first n terms  2 ( 1) ( )22 n nnS a n d a L= + − = + where last term, ( )1L a n d= + − . (1 ) ( 1) 11 nn n a r a rS rr −−== −− , where 1r . Sum to infinity Does not exist 1 aS r  = − , S exists if 11 − r Sequences Examples Defined by a formula for the nth term Defined by a relation 1 f( )+ =nnuu { 1, 3, 5, 7, ...} f ( ) 2 1nn=− , 1n 11 1, 2 for 1nnu u u n += = +  {4, 8, 16, 32, ...} 1f ( ) 2 nn += , 1n 11 4, 2 for 1nnu u u n += =  1 2 3 4, , , , ...2 3 4 5  f ( ) 1 nn n= + , 1n  2, 5, 2, 5, ...−− 11 2, 3 for 1nnu u u n +=− =− +  Series and the Sigma Notation Using the notations Sn and nu Using the sigma notation Sum of the first n terms 1 2 3nnS u u u u= + + + + n n r r uuuuu ++++= = 321 1 nth term 1n n nu S S −=−  − == −= 1 11 n r r n r rn uuu First term 11uS=  = = 1 1 1 r ruu Sum from the thm term to the thn term where nm 12 1 m m m n nm u u u u SS ++ − + + + =− 1 11 12 n n m r r r r m r r m m m n u u u u u u u − = = = ++ =− = + + +   

Anglo-Chinese Junior College 2025 H2 Mathematics 9758: Sequences & Series / Summary / Page 2 of 3 Convergence A sequence  nu is convergent  as nu l n→ → , where 𝑙 is a finite number. i.e. lim n n ul → = , a finite number. Otherwise, it is divergent. A series nS is convergent  as nS l n→ → , where 𝑙 is a finite number. i.e. lim n n Sl → = , a finite number. Otherwise, it is divergent. Common series that can be evaluated immediately Series Examples 1 ( ) 1 ... 1 n rm n m times k k k k k n m k = −+ = + + + + = − + , where k is a constant. 26 5 26 5 1 22 3 3 3 3 ... 3 22 3 66 r times= − + = = + + + + =  =  2 ( ) ( )1 2 ... n rm r m m m n = = + + + + + + (AP)  1 2 nm mn−+=+ ( ) 20 3 3 4 5 ... 20 18 3 202 r r = = + + + + =+  3 12 ... n r m m m n rm k k k k k ++ = = + + + + (GP) ( )11 1 nmmkk k −+−= − 14 5 6 7 14 5 5 10 2 2 2 2 ... 2 2 (1 2 ) 12 r r= = + + + + −= −  Some series, e. g. 2 1 n r r =  , 3 1 n r r =  have standard results that will be given should a question require their use. Series that require formula or algebraic manipulation to evaluate Suppose it is given (or found earlier in a question) that ( ) ( ) ( )1 1 3 1 1 2 4 2 1 2 2 n r r r n n= = − −+ + + …………..(*) 1 Change of upper index Find (a) ( ) 2 1 1 2 n r rr + = + and (b) ( )1 1 2r rr  = + . ( ) ( ) ( ) ( ) ( ) 2 1 1 3 1 1 24 2 2 1 2 2 2 3 1 1 .4 2 3 2 4 n r rr nn nn + = = − −+ + + + +       = − − ++ (a) (b) As n→ , ( ) 1 021n →+ and ( ) 1 022n →+ . ( ) ( ) 3 1 1 3 4 2 1 2 2 4 − − →++nn Hence ( )1 13 24 n r rr= →+ as n→ , i.e. ( )1 13 24r rr  = =+ . Substitute n in (*) by 2n+

Anglo-Chinese Junior College 2025 H2 Mathematics 9758: Sequences & Series / Summary / Page 3 of 3 2 Change of lower index Find ( )5 1 2 n r rr= + . ( ) ( ) ( ) 4 5 1 1 1 1 1 2 2 2 nn r r r r r r r r r= = = =−+ + +   ( ) ( ) ( ) ( ) ( ) ( ) 3 1 1 3 1 1 4 2 1 2 2 4 2 4 1 2 4 2 11 1 1 .60 2 1 2 2 nn nn = − − − − − + + + +  = − − ++ 3 Change of f (r) and index by substitution In general, ( ) ( ) ( ) ( ) 1 10 1 2 2 3 f f 1 f1 f2 etc. nn rr n r n r rr r r − == + = + = =+ =− =−    The above summations are all equivalent. To find ( ) 3 4 1 2 n r rr + = − , first express what you want to find (i.e. ( ) 3 4 1 2 n r rr + = − ), in terms of the result for the function you already have: ( ) ( )( ) ( ) ( ) 3 2 3 4 2 4 1 2 1 2 11 2 2 2 2 1 2 1 .2 n r n rr n r n r r r r r rr rr + + = + = + = + = + = =− + + − = + = +    Then evaluate the series using 1 and/or 2 above. [Ans: ( ) ( ) 5 11 12 2 2 2 3nn++−− ] replace r by 1r+ replace r by 1r− replace r by 2r− replace r by 2r+