Seq Series APQ Soln (RVHS)

Ch 2 Sequences and Series (Solutions to Tutorial & Assignment) RIVER VALLEY HIGH SCHOOL 6 Sequence & Series Part I: Additional Practice Questions 1. Given that θ is sufficiently small, show that tan 3 4 3 π θ θ   + ≈ +     . ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) 1 2 tan 3 tan tan 3 1 tan tan 3 3 tan 1 3 tan 3 1 3 3 1 3 3 1 1 3 3 1 3 3 3 3 3 4 (shown) π θ π θ π θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ −   +    + = − += − +≈ − = + − ≈ + + − − = + + = + + + ≈ + 2. [MI/2020/Promo/PU2/P1/Q6(b)] Find the expansion of 2 1 3 9 x x + − in ascending powers of x, up to and including the term in 3x . State the set of values of x for which the expansion is valid. [4] ( ) ( ) ( ) ( ) ( ) 1 2 2 2 1 2 2 1 1 2 2 2 1 2 2 1 3 1 3 9 9 1 3 9 1 9 1 3 9 1 9 1 1 3 1 3 9 x x x x xx xx xx − − − − − + = + − −     = + −               = + −             = + −    

Ch 2 Sequences and Series (Solutions to Tutorial & Assignment) RIVER VALLEY HIGH SCHOOL 7 ( ) ( ) 2 2 2 3 2 3 1 1 1 1 3 1 ... 3 2 9 1 1 1 3 1 ... 3 18 1 1 1 1 3 ... 3 18 6 1 1 1 ... 3 54 18 x x x x x x x x x x       = + + − − +               = + + +       = + + + +     = + + + + For expansion to be valid, 2 2 19 9 3 3 x x x − < < ∴ − < < 3. [TJC/2020/J1MYE/1] Expand 2 1 2 2 x x −    −  in ascending powers of x up to and including the term in 2x and state the range of values of x for which this expansion is valid. [4] ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 1 2 (1 2 ) ( 2) 2 1 4 4 ( 2) 1 2 1 1 4 4 1 3 4 2 1 3 1 4 4 4 4 4 1 3 3 4 4 16 x x x x xx x xx x x x x x x x x x − − − −  = − −   −    = − + − −         = − + + + +             = + + − − + +     = − + + ⋯ ⋯ ⋯ Expansion is valid for 12 x < , i.e. 2 2 x− < <

Ch 2 Sequences and Series (Solutions to Tutorial & Assignment) RIVER VALLEY HIGH SCHOOL 33 Sequence & Series Part II: Additional Practice Questions 1. An arithmetic progression has 100 terms and first t erm a. If the sum of the last 50 terms is 1875 more than the sum of the first 50 terms, fi nd the common difference. If 2, a = find the least value of n such that the sum of the first n terms of the arithmetic progression exceeds 122.

Ch 2 Sequences and Series (Solutions to Tutorial & Assignment) RIVER VALLEY HIGH SCHOOL 34 2. [N2009/I/8] Two musical instruments, A and B, consist of metal bars of decreasing lengths. (i) The first bar of instrument A has length 20cm and the lengths of the bars form a geometric progression. The 25th bar has length 5cm. Show that the total length of all the bars must be less than 357cm, no matter how many bars there are. [4] Instrument B consists of only 25 bars which are identical to th e first 25 bars of instrument A. (ii) Find the total length, L cm, of all the bars of instrument B and the length of the 13th bar. [3] (iii) Unfortunately the manufacturer misunderstands the i nstructions and constructs instrument B wrongly, so that the lengths of the bars are in ar ithmetic progression with common difference d cm. If the total length of the 25 bars is still L cm and the length of the 25th bar is still 5cm, find the v alue of d and the length of the longest bar. [4] (i) ar 24 = 20 r24 = 5 r24 = 1 4 r = 12 1 2 Total length of all bars < S ∞ = 20 1 – 12 1 2 = 356.343 < 357 (shown) (ii) L = 20 1 –    12 1 2 25 1 – 12 1 2 = 272.2573 272 ≈ Length of 13th bar = 20    12 1 2 12 = 20 × 1 2 = 10 cm (iii) Let b = length of first bar of instrument B. 25 2 [ 2 b + 24 d ] = L b + 12 d = L 25 -----(1) Also b + (25–1) d = 5 24 5 b d  + = -----(2) (2) – (1): So 12 d = 5 – L 25 = 5 – 272.2573 25 ∴ d = –0.49086 0.491 ≈ − Length of longest bar = b = 5 – 24(–0.49086) = 16.8 cm

Ch 2 Sequences and Series (Solutions to Tutorial & Assignment) RIVER VALLEY HIGH SCHOOL 35 3. The positive multiples of 3 are grouped into sets as { } { }1 2 3 , 6,9 , A A = = { }3 12,15,18 A = and { }4 21, 24, 27,30 ,... A = where the set nA contains n elements. (i) Find the total number of elements in the first n sets. Show that the last element of the set nA is given by 3 ( 1). 2 n n + (ii) Hence, or otherwise, find the sum of all the elements in the set nA in terms of n.

Ch 2 Sequences and Series (Solutions to Tutorial & Assignment) RIVER VALLEY HIGH SCHOOL 36 4. [N2011/I/9] (i) A company is drilling for oil. Using machine A, the depth drilled on the first day is 256 metres. On each subsequent day, the depth dr illed is 7 metres less than on the previous day. Drilling continues daily up to an including the day when a depth of less than 10 metres is drilled. What depth is drilled on the 10th day, and what is the total depth when drilling is completed? [6] (ii) Using machine B, the depth drilled on the first day is also 256 me tres. On each subsequent day, the depth drilled is 8 9 of the depth drilled on the previous day. How many days does it take for the depth drilled to exceed 99% of the theoretical maximum total depth? [4] Solutions: (i) 256 a = , 7d = − Depth drilled on 10 th day = ( )10 10 1 T a d = + − ( )( )256 9 7 193 metres = + − = Want ( )1 10 nT a n d = + − < ( )( )256 1 7 10 256 7 7 10 7 253 253 1 36 7 7 n n n n  + − − <  − + <  >  > = So it takes 37 days for drilling to complete. Total depth ( ) ( )( )37 37 2 256 37 1 7 4810 metres 2S   = + − − =   (ii) 256 a = , 8 9r = , Theoretical maximum total depth 1 aS r∞ = − Want ( )1 0.99 1 n n a r S S r ∞ − = > − ( )1 0.99 1 1 1 0.99 8 0.01 9 ln 0.01 39.09875612 8ln 9 n n n a r a r r r n n −    >   − −    − >    <     >  > Thus, it will take 40 days

Ch 2 Sequences and Series (Solutions to Tutorial & Assignment) RIVER VALLEY HIGH SCHOOL 37 5. *In the triangle ABC , 8, 10 and 6. AB BC CA = = = AD is perpendicular to , BC DE to AB , EF to BD , FG to EB and so on. Find the sum to infinity of ... CA AD DE EF FG + + + + + G E D C F B A

Ch 2 Sequences and Series (Solutions to Tutorial & Assignment) RIVER VALLEY HIGH SCHOOL 38 6. [HJC03/1/1] By writing ( ) ! 1 323 2 + −− r rr in the form ( ) ( ) ! 1!! 1 +++− r C r B r A , find ( ) = + −−n r r rr 1 2 ! 1 323 . [4]

Ch 2 Sequences and Series (Solutions to Tutorial & Assignment) RIVER VALLEY HIGH SCHOOL 39 7.* [SAJC03/1/4 (modified)] Given the series ... 4 5log 83 4log 62 3log 41 2log 2 +     +     +     +     . Let Sn denote the sum of the first n terms of the above series. Write down the values of Sn for n = 1, 2 and 3 . Suggest an expression for Sn for all positive integers n.

Ch 2 Sequences and Series (Solutions to Tutorial & Assignment) RIVER VALLEY HIGH SCHOOL 40 8. (a) Find the sum of the series 3 5 5 7 7 9 ... ⋅ + ⋅ + ⋅ + n terms. (b) Find the sum of 22222 100 .... 4321 ++−+−=S . (c) Given that 2 3 2 1 ( 1) 4 n r nr n = = +  show that  = −− n n r nr 2 3 ) 13 ( 21 ( 1) 9 (5 1) 4( 1) 4 n n n n   = + + − +   (a) ( )( )3 5 5 7 7 9 ... 2 1 2 3 n n ⋅ + ⋅ + ⋅ + + + +  = ++= n r rr 1 ) 32 )( 12 (  = ++= n r rr 1 2 ) 384 (  === ++= n r n r n r rr 111 2 384 nnnnnn 3) 1(2. 8) 12 )( 1(6. 4 +++++= nnnn 3] 6) 12 ( [ ) 1(3 2 ++++= nnnn 3) 72 )( 1(3 2 +++= [ ]9) 72 )( 1( 23 1 +++= nnn 21 (4 18 23) 3 n n n = + + (b) ( ) ( )22222222 100 ... 64299 ... 531 ++++−++++=S  == −−= 50 1 2 50 1 2 ) 2 () 12 ( rr rr  == −+−= 50 1 2 50 1 2 4) 144 (