Qm- Session #05- Logarithm, Progressions , Functions

Basics of Mathematics Key concepts : Logarithms, Progressions – Functions

The logarithm to the base b of the variable x is defined as the power to which you would raise b to get x. If the logarithm to the base b of x is equal to y, then b raised to the y power will give you the value x. It it written x = by is the same as y = logbx

Here is a riddle for you. If we have a full binary tree (no branches are cut off) that has 8 smallest branches, how many "levels of branching in two" are there? Let us see. The last branching happens when 4 branches became 8:

The branching before that happens when 2 branches become 4:

and the very first one is when a single branch becomes 2

So we have three levels of branching

The general combination properties of logarithms are: logb1 = 0 logbb = 1 logbbx = x

logb(xy) = logbx + logby logb(x/y) = logbx – logby logb(xy) = y logbx

Example 1: 1000 = 103 is the same as 3 = log101000. Example 2: log381 = ? Example 2: log5125 = ?

Arithmetic Progression A sequence is called an arithmetic progression if the difference of any term from its preceding term is constant. This constant is usually denoted by d and is called common difference. General term of an A.P. Tn = a +(n -1) d.

Formulae for Sum of A.P. Sum of first n terms of an A.P. is

n S n = [ 2a + ( n − 1) d ] 2

If Sn is sum of n terms of an A.P. whose first term is a and last term is l,then

n Sn = ( a + l ) 2

If the 9 term of an th A.P. is 99 and 99 th term is 9, find its 108 term. th

Find the sum of the series 1 +3 +5 +... +99.

How many terms of the A.P. 17 + 15 + 13 + ... must be taken so that sum is 72?

Geometric Progression (G.P.) A sequence is called a geometric progression if the ratio of any term to its preceding term is constant. This non-zero constant is usually denoted by r and is called common ratio. Ex. 9 , 18 , 36 , 72…….

General term of a G. P. Tn = a r n-1

Find the next term of the sequence 1/6, 1/3, 2/3...

Harmonic Progression A series is said to be harmonic progression if the series obtained by taking reciprocals of the corresponding terms of the given series is an arithmetic progression. For example

1 1 1 1, , , ,...... 4 7 10

• A general H.P. is

1 1 1 1 , , , ,..... a ( a + d ) ( a + 2d ) ( a + 3d ) • nth term of an H.P.

1 [a +(n −1)d ]

The 7 term of an H.P. th is 1/10 and 12 term is th 1/25 Find the 20 term th

Functions • Indicates relationship among objects • Expresses the relationship of one variable or a group of variables (domain) with another variable (range) by associating every member in the domain to a unique member in range. y= f(x)

•

y= f(x,z)

1 dimensional function 2 dimensional function

Types of Function • • • •

Linear function Inverse function Exponential function Logarithmic function

Linear function y = mx + c y - dependent variable x – independent variable m – slope c - intercept

Inverse function

y = f ( x) x=f

−1

( y)

Exponential function

y = ma a – base

x

Logarithmic function

If x = b then y = log b x y

b = base