Arithmetic and Geometric Sequences and their Summation

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1 4 Arithmetic and Geometric Sequences and their Summation O-FOUDATIO 4F 4.4 Series 4.5 Arithmetic Series ame : Date : Mark : Key Concepts and Formulae Sum of the first n terms of an arithmetic series: na ( + l) n Sn ( ) Sn ( ) [ a+ ( n- ) d] or where a and l are the first term and the last term of the series, with common difference d.. Find the sum of the terms of the following arithmetic series to 0 terms to terms (c) 3 + (x + 6) + (x + 9) + to 5 terms Q a ( ) and d ( 4 ) - ( ) ( ) \ S( 0) 0 [ ( ) + (0 -)( )] ( 0 ) Q a ( 56 ) and d ( 5 ) - ( 56 ) ( -5 ) \ S() [ 56 ( ) + ( - )( -5)] 77 (c) Q a 3 and d (x + 6) - 3 x + 3 \ 5 S( 5) [() 3 + ( 5 - )( x + 3)] 5( 7x + 4) 9

2 4 Arithmetic and Geometric Sequences and their Summation. Find the number of terms and the sum of the terms of the following arithmetic series º + 7 Let a, d and n be the first term, the common difference and the number of terms of the given series respectively. \ a ( -3 ), d - ( -3 ) ( 4 ), l T(n ) 65 and Tn ( ) a+ ( n-) d \ 65 ( -3 ) + ( n -)( 4 ) 6 4n - 4 n ( ) \ The arithmetic series has ( ) terms. ( )[( -3 ) + 65] S( ) ( 55 ) \ The sum of the terms of the arithmetic series is ( 55 ). Let a, d and n be the first term, the common difference and the number of terms of the given series respectively. Q a 4, d 3-4, l Tn ( ) 7 and T(n ) a + (n - )d \ 7 5 n + ( n - ) Ê ˆ 4 Ë n - 6 \ The arithmetic series has 6 terms. S( 6) ( ) \ The sum of the terms of the arithmetic series is

3 umber and Algebra 3. How many terms must be taken from each of the following arithmetic series in order to obtain a sum as indicated? (-6) + (-3) º Let a, d and be the first term, the common difference and the number of terms that must be taken respectively. Q a ( -6 ), d ( -3 ) - ( -6 ) ( 3 ) and S( ) 99 and S( ) [ a + ( -) d] \ 99 [ ( -6 ) + ( -)( 3 )] 99 ( 3-5) ( - )( + 6) 0 or -6 (rejected) \ ( ) terms of the arithmetic series must be taken. Let a, d and be the first term, the common difference and the number of terms that must be taken respectively. Q a 0 4, d and S( ) 75 and S ( ) [ a + ( - ) d] \ 5 75 Ê Ë ( - 5)( 3-40) È Ê ˆ 0 ( ) Ë Ê ˆ Ë - ÎÍ 4 5 or ˆ 4 (rejected) \ 5 terms of the arithmetic series must be taken. 00

4 4 Arithmetic and Geometric Sequences and their Summation 4. Find the common difference of an arithmetic series if its first term is -6 and the sum of the first 0 terms is -70. Let a, d and S(n) be the first term, the common difference and the sum of the first n terms of the arithmetic series. Q a ( -6 ) and S(0) ( -70 ) 0 \ ( -70 ) ( ) + - [ -6 ( 0 ) d] ( 9d - 3) - 4 9d - 3 9d d ( ) \ The common difference of the arithmetic series is ( ). 5. Sarah plans to finish reading a novel of 50 pages within two weeks. She reads 5 pages on the first day, and on each subsequent day she reads 4 pages more than what she read on the preceding day. Can she finish reading the novel within two weeks? The numbers of pages Sarah reads on consecutive days form an arithmetic sequence with the first term 5 and the common difference 4. Let S(n) be the total number of pages Sarah reads on the first n days. 4 \ S( 4) [ ( 5) + ( 4 - )( 4)] 574 < 50 \ Sarah cannot finish reading the novel within two weeks. 0

5 umber and Algebra 6. Raymond s starting salary with a company is $5 000 per month and he has a constant annual increment of $00 per month. When he leaves the company, his salary is 3 times that of his starting salary. Find the total number of years he has worked for the company, the total amount of salary he has earned. His monthly salaries for consecutive years form an arithmetic sequence with the first term $5 000 and the common difference $00. Let n and $T(n) be the total number of years he has worked for the company and his monthly salary after working for n years. Q Tn ( ) \ ( n -)( 00) ( n -) n - 5 n 6 \ He has worked for the company for 6 years. His starting annual salary $5 000 $0 000 Annual increment $00 $4 400 Let $S(n) be the total amount of salary he earned in the first n years. 6 S( 6) [ ( 0 000) + ( 6 - )( 4 400)] \ The total amount of salary he has earned is $

6 4 Arithmetic and Geometric Sequences and their Summation 7. Find the sum of all the non-negative terms of the arithmetic series º Let k be the number of non-negative terms of the arithmetic series. The kth term T(k) will then be the smallest non-negative term. Let a and d be the first term and the common difference of the arithmetic series respectively. 5 Q a 4, d and T(k) 0 \ T(k) ( k - ) ˆ Ë k k 53 k 53 5 Q k is the number of terms, it must be an integer. \ k 0 The required sum 0 È + - Ê 5 4 ( ) ( 0 ) ˆ ÎÍ Ë Given that log 3 a, express log 3 + log 3 + log log 3 55 in terms of a log 3 + log 3 + log 3 + º + log log( º 3 ) log 3 log 3 log 3 55( + 55) 540 log 3 540a º

7 umber and Algebra 9. Find the sum of the integers between 00 and 400 inclusive that are divisible by 4, 9, (c) both 4 and 9, (d) neither 4 nor 9. The integers between 00 and 400 inclusive that are divisible by 4 form the arithmetic sequence 00, 04, 0,, 400. Let a, l and n be the first term, the last term and the number of terms of the arithmetic series respectively Q a 00, l 400 and n ( ) \ The required sum The integers between 00 and 400 inclusive that are divisible by 9 form the arithmetic sequence 0, 7,, 396. Let a, l and n be the first term, the last term and the number of terms of the arithmetic series respectively Q a 0, l 396 and n \ The required sum 33( ) 36 04

8 4 Arithmetic and Geometric Sequences and their Summation (c) Integers that are divisible by both 4 and 9 are divisible by 36. The integers between 00 and 400 inclusive that are divisible by both 4 and 9 form the arithmetic sequence 0, 44,, 396. Let a, l and n be the first term, the last term and the number of terms of the arithmetic series respectively Q a 0, l 396 and n \ The required sum 9( ) 6 (d) By and (c), the sum of the integers between 00 and 400 inclusive that are divisible by 4 but not \ the sum of the integers that are divisible by either 4 or The sum of the integers that are divisible by neither 4 nor 9 the sum of the integers between 00 and 400 inclusive - the sum of the integers between 00 and 400 inclusive that are divisible by either 4 or 9 30 ( )

tiple Choice tify the letter of the choice that best completes the statement or answers the question.

T PREP tiple Choice tify the letter of the choice that best completes the statement or answers the question. 1. The owner of a small business makes $76,000 per year. His business manager makes $49,000.