Now, in this last section, we will look at all the miscellaneous series problems based on Arithmetico-geometric series, Iterative series, User-defined series and others.
Type I: Arithmetico-Geometric Series (AGP)
E.g. 1: Find the sum to infinite terms:
(y + 4y2 + 7y3 + 10y4 + 13y5 + ……………………)
The series given above is an AGP. We can easily identify an AGP as one term of the series will be in AP while the other one will be in GP.
In this series y, y2, y3, y4, y5 are in GP with a common ratio of ‘y’, while the terms 1, 4, 7, 10, 13 and so on are in AP.
So, the standard way of solving AGP problems is to multiply the entire series by the common ratio of the GP and then subtract the newly formed series from the original series. Why do we subtract?
Because on subtracting, we will get a pure geometric progression of infinite terms. And, we know how to find the sum of infinite terms of a GP.
After learning the theory of arithmetic progression, we will see the basic fundas and applications of Geometric Progression. In AP, we used to get next term by adding a common difference to the previous term. But, in case of GP, we can obtain a new term by multiplying the previous term by a common ratio.
The first term is usually denoted by ‘a’ and the common ratio by ‘r’.
For e.g., 2, 4, 8, 16, 32,……………….is an example of GP with first term being 2 and common ratio being 2.
We can also say that 27, 9, 3, 1,…………………..is also an example of GP with a = 27 and r = 1/3.
As we found out the nth term of AP, we can also find out nth term of GP by the same logic.
So, t1 = a
Then, t2 = a × r, t3 = a × r × r or ar2, t4 = ar3, t5 = ar4,…………………..tn = a × r(n – 1)
So, we can use this formula whenever question is related with last term, first term or number of terms.
While solving problems, we generally come across terms which are in arithmetic progression, but we are not able to identify it. We can find the sum of terms, number of terms of the series if familiar with the concepts of AP.
For example, 3, 5, 7, 9, 11, 13,………is an example of an Increasing arithmetic progression with the common difference of 2. Similarly, 3, 0, –3, –6, –9,………is also an example of arithmetic progression with common difference of (–3). So, when the common difference is positive, the AP is increasing while the AP is decreasing when the common difference is negative.
We usually denote the first term by ‘a’ and if the common difference is taken as‘d’, the second term would be (a + d), third term would be (a + d + d) or ‘a + 2d’ and so on we can find the following terms.
The most important thing in Absolute value (Modulus) is its definition. Without its definition, modulus will not exist. In other words, the definition of Modulus is sacrosanct. Absolute value or Modulus is denoted by || sign.
|x|= x if x ≥ 0.
And |x|= –x, if x < 0.
So, the definition of modulus says that if any number is positive, let’s say ‘x’, then ‘x’ will come out of the modulus as it is.
While if ‘x’ is negative, then ‘x’ will come out of the modulus with a negative sign and it will become –x.
For example, |5| = 5 because number inside modulus is 5 which is a positive number, so when it comes out of mod symbol, it will remain as it is.
Another example: |–1.5|= 1.5, because number inside mod symbol is –1.5 which is a negative number, so when it comes out of mod symbol, it will become –(–1.5) = 1.5.
So, we can conclude that any number coming out of modulus symbol will be positive.
First of all, we will learn how to plot the graph of this. Once, we know how to plot, we can find out the area and perimeter orally without plotting the graph.
If there is a single modulus involved in expression, we get two equations. But if two mods are involved, we will get four equations under different conditions. In this case also, we will get 4 equations.
1st equation: x + y = 5 if x ≥ 0 and y ≥ 0.
2nd equation: x – y = 5 if x ≥ 0 and y < 0.
3rd equation: –x + y = 5 if x < 0 and y ≥ 0.
4th equation: –x – y = 5 if x < 0 and y < 0.
So, we get four different equations with different conditions. So we need to plot these lines and use the conditions in which quadrant x and y is positive or negative.
There are many ways of finding the area of the required region. We will look at two of those methods and both those methods are pretty easy.
What is basic difference between solving an Inequality and Equality problem?
Answer is pretty simple. While solving an equality problem, we get a fixed value/s of the variable but in case of an inequality, we get a range of values.
For e.g., On solving 2x – 5 = x – 3, we get value of the variable x as 2.
But the same problem with inequality will give us range of values. 2x – 5 > x – 3; On solving this, we get x > 2. The range signifies that ‘x’ can take all the values which are greater than 2 and uptill + infinity.
Let us see pictorial understanding of ‘>’ and ‘<’.
Standard equation of a Quadratic Polynomial is ax2 + bx + c, and since the greatest degree of the polynomial is 2, it can have maximum of two roots. We will discuss all the possibilities of quadratic equation having two roots, one root or no root.
The graph of a quadratic polynomial depends on the coefficient of x2 i.e. ‘a’. If ‘a’ is positive, graph of quadratic polynomial will be a upright parabola (U-shaped) and if ‘a’ is negative, graph will be of the shape of inverted parabola. The logic is same as that of a linear polynomial as the coefficient of ‘x’ (y = mx + c) determines whether the line will be increasing or decreasing.
Let us plot a quadratic polynomial y = x2 – 3x – 4; While plotting the graph we should take care of few points. First of all, we should find out the roots of the polynomial as the roots will help us in determining the points where the graph will cut the X-axis. And secondly, we can also find out the y-intercept easily which will be constant part in the equation given. In this equation given, roots are (–1 &4) and y-intercept will be –4. Also, the graph of a quadratic polynomial will increase rapidly when we increase the value of x or decrease the value of x.
On observing the table, x = –1 and x =4 are roots of the polynomial as value of y-coordinate is equal to 0. Also, y-intercept is –4 as the value of x-coordinate is equal to 0. And also as the value of ‘x’ starts increasing, value of ‘y’ increases rapidly. And when value of ‘x’ decreases, then also ‘y’ increases rapidly.