IB_Mathematics_HL_Revision

IB_Mathematics_HL_Revision - Rahul Chacko IB Mathematics HL...

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Rahul Chacko IB Mathematics HL Revision – Step One Chapter 1.1 – Arithmetic sequences and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series. Sigma notation. Arithmetic Sequences Definition: An arithmetic sequence is a sequence in which each term differs from the previous one by the same fixed number: { u n } is arithmetic if and only if d u u n n 1 . Information Booklet  d n u u n 1 1 Proof/Derivation: d u u n n 1 d u u n n 1 dn u u n 1 1 dn u u n 1 d n u u n 1 1 Derivations: d n u u n 1 1 1 1 n u u d n 1 1 d u u n n Information Booklet n n u u n d n u n S 1 1 2 1 2 2 Proof: S n = u 1 + u 2 + u 3 + …+ u n = u 1 + ( u 1 + d ) + ( u 1 + 2 d ) + ( u 1 + 3 d ) + …+ ( u 1 + ( n 1) d ) = u n + ( u n d ) + ( u n 2 d ) + ( u n + 3 d ) + …+ ( u n ( n 1) d ) 2 S n = n ( u 1 + u n ) n n u u n S 1 2 Derivations
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1 2 u n S u n n n n u n S u 2 1 n n u u S n 1 2 Geometric Sequences Definition: A geometric sequence is a sequence in which each term can be obtained from the previous one by multiplying by the same non-zero constant. { u n } is geometric if and only if , 1 r u u n n n where r is a constant. Information Booklet 1 1 n n r u u Proof: r u u n n 1 r u u n n 1 n n r u u 1 1 1 1 n n r u u Derivations: 1 1 n n r u u 1 1 1 n n u u r 1 log 1 u u n n r (non-calculator paper) 1 log log 1 r u u n n (calculator paper) Compound Interest: n n r u u 1 1 , where 1 u initial investment, % 100 % % 100 i r , i interest rate per compounding period, n = number of periods and 1 n u amount after n periods. 1 n n u r u
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Information Booklet     r r u r r u S n n n 1 1 1 1 1 1 , r 1 Proof: S n = u 1 + u 2 + u 3 + + u n-1 + u n = u 1 + u 1 r + u 1 r 2 + u 1 r 3 + … + u 1 r n 2 + u 1 r n 1 rS n = ( u 1 r + u 1 r 2 + u 1 r 2 + u 1 r 3 + u 1 r 4 + … + u 1 r n 1 ) + u 1 r n rS n = ( S n u 1 ) + u 1 r n rS n S n = u 1 r n u 1 S n ( r 1) = u 1 ( r n 1)    1 1 1 r r u S n n Derivations 1 1 1 n n r r S u r u S r n n log 1 log 1 1 2 3 3 2 1 ... 1 1 1 n n n n n r r r r r r r r u S Sum to infinity r u S 1 1 , 1 r Proof:   r r u S n n 1 1 1 , 1 r , 0 r , 1 r r u S 1 0 1 1 , 1 r r u S 1 1 , 1 r Sigma Notation n r n f 1 ) (
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n is the number of terms, f ( n ) is the general term and r = the first n value in the sequence.
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This note was uploaded on 05/06/2010 for the course MATH 1102 taught by Professor Chang during the Winter '10 term at Savannah State.

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IB_Mathematics_HL_Revision - Rahul Chacko IB Mathematics HL...

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