SPECIAL HL GRADE SUMS

SEQUENCES AND SERIES

FROM: SRIRAMAN .IYER. HEAD, DEPT OF MATHEMATICS IB PROGRAMS, MUMBAI .INDIA EMAIL:ib.mathfaculty@gmail.com

(1) if 3, log y x, 3 log zy, 7 logxz are in

Arithmetic Sequence

Prove that

(a) X 18 = y21 = z28

(b) Find x in terms of Y and Z separately

(2) If the ratio of Sum of first m terms and

Sum of first n terms of an Arithmetic

series is given by

(𝑚

𝑛)2

(a) Find the ratio of the mth term and

nth term

(b) Prove that the ratio never contains

Even Numerator and Denominator for

any value of m and n, where m and n

are positive integers

(c).if u = f(m) and v = f(n) , then find the

Derivative of u and v with respect to its

independent variables

(3) If the (m+1) th , (n+1) th and ( (r+1) th

terms of an ARITHMETIC SEQUENCE are in

GEOMETRIC SEQUENCE.

a. Find the ratio between common

difference and first term

b. Prove that the above ratio is

negative for any value of n if 2𝑚𝑟

𝑚+𝑟 = n

(4) A sequence is given by

2+5+12+31+86+………..

(a) Find the nth term of the above

sequence

(b) Find the sum to n terms of the above

sequence

(c) Sn = f(n) , then find S’(n)

(d) Find the sum till 20 terms

(5) (a) Find the ratio of Sum of n terms of

sequences of “Sum of Natural numbers”

and “Sum of Squared Natural Numbers”

(b). Find the ratio of (a) for the first 10

terms

(c) if f(n) represents the ratio (a), and g(n)

represents the “ Sum of cubes of first n

natural numbers” , then find the

composite function of f and g

(d) Check the commutative law of

composite function of f and g

(6) If Y = x+ x3+x5+….. is a infinite Geometric

sequence , find

a. Sum of first 10 terms for x>1

b. Find the value of Y

c. If Y = f(x) , find f’(x)

d. Find the f-1(x)

(7) If f(x) = 𝑖 log 𝑥𝑛𝑖=1

a. Find f(x).

b.Find f(x) at x=10

c.Find f(x) at x=n=10

d.Find the inverse function of f(x)

e.Find f’(1) at n=100

(8) Find the sum of n terms of 𝑥

1−(𝑥2) +

(𝑥2)

1−(𝑥4) +

(𝑥4)

1−(𝑥8) + ……….

(a) Find the nth term

(b) Find the sum of first n terms

(c) Find the sum when n = 4

(d) If tn= f(n), find f’(1)

(9) In an Arithmetic Sequence, if m.tm=n.tn

a.Prove that t(m+n)=0

b.Find the ratio of 5th term and 10th term

(10) A Series is given by

7+77+777+……. = S

a.Find the nth term of the Series

b.Find S for the first n terms

c.Find S for first 20 terms using (a)

d.If S = f(n), then find f’(n)

d.find the n