MEI Structured Mathematics: Formulae and Tables

MEI STRUCTURED MATHEMATICS

EXAMINATION FORMULAE AND TABLES

1

Arithmetic series
General (kth) term, last (nth) term, l =
Sum to n terms,
Geometric series
General (kth) term,
Sum to n terms,
Sum to infinity

Infinite series f(x) uk = a + (k – 1)d un = a + (n – l)d
–
–
Sn = 1 n(a + l) = 1 n[2a + (n – 1)d]
2
2

x2 xr = f(0) + xf'(0) + –– f"(0) + ... + –– f (r)(0) + ...
2!
r!

f(x) f(a + x)

uk = a r k–1 a(1 – r n) a(r n – 1)
Sn = –––––––– = ––––––––
1–r
r–1

(x – a)2
(x – a)rf(r)(a)
= f(a) + (x – a)f'(a) + –––––– f"(a) + ... + –––––––––– + ... r! 2! x2 xr
= f(a) + xf'(a) + –– f"(a) + ... + –– f(r)(a) + ...
2!
r!

x2 xr ex = exp(x) = 1 + x + –– + ... + –– + ... , all x
2!
r!

a
S∞ = ––––– , – 1 < r < 1
1–r

x2 x3 xr
= x – –– + –– – ... + (–1)r+1 –– + ... , – 1 < x р 1
2
3 r sin x

x3 x5 x 2r+1
= x – –– + –– – ... + (–1)r –––––––– + ... , all x
3!
5!
(2r + 1)!

cos x

x2 x4 x 2r
= 1 – –– + –– – ... + (–1)r –––– + ... , all x
2!
4!
(2r)!

arctan x

x3 x5 x 2r+1
= x – –– + –– – ... + (–1)r –––––– + ... , – 1 р x р 1
3
5
2r + 1

General case

sinh x

n(n – 1) n(n – 1) ... (n – r + 1)
(1 + x)n = 1 + nx + ––––––– x2 + ... + ––––––––––––––––– xr + ... , |x| < 1,
2!
1.2 ... r n∈‫ޒ‬ x3 x5 x 2r+1
= x + –– + –– + ... + –––––––– + ... , all x
3!
5!
(2r + 1)!

cosh x

x2 x4 x 2r
= 1 + –– + –– + ... + –––– + ... , all x
2!
4!
(2r)!

artanh x

x3 x5 x 2r+1
= x + –– + –– + ... + –––––––– + ... , – 1 < x < 1
3
5
(2r + 1)

Binomial expansions
When n is a positive integer n n n (a + b)n = an + 1 an –1 b + 2 an–2 b2 + ... + r an–r br + ... bn , n ∈ ‫ގ‬ where n n n n+1 n! n r = Cr = –––––––– r + r+1 = r+1 r!(n – r)!

()

()

()

()

() ( ) (

)

2
Logarithms and exponentials exln a = ax

logbx loga x = ––––– logba Numerical solution of equations f(xn) Newton-Raphson iterative formula for solving f(x) = 0,