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)