We are offering TS 10th Class Maths Notes Chapter 6 Progressions to learn maths more effectively.

## TS 10th Class Maths Notes Chapter 6 Progressions

→ The array of numbers following some rule is called a number pattern.

E.g.: 4, 6, 4, 6, 4, 6,…

→ There is a relationship between the numbers of a pattern.

→ Each number in a pattern is called a term.

→ The series or list of numbers formed by adding or subtracting a fixed number to / from the preceding terms is called an Arithmetic Progression, simply A.P.

E.g.: 3, 5, 7, 9, 11,

→ In the above list, each term is obtained by adding ‘2’ to the preceding term except the first term.

→ Also, we find that the difference between any two successive terms is the same throughout the series. This is called “common difference”.

→ The general form of an A.P. is a – the first term; d – the common difference.

a, a + d, a + 2d, a + 3d,…. a + (n – l)d. Here d = a_{2} – a_{1} = a_{3} – a_{2} = a_{4} – a_{3} = ………………… = a_{n} – a_{n-1}.

→ If the number of terms of an A.P. is finite, then it is a finite A.P.

E.g.: 10, 8, 6, 4, 2.

→ If the number of terms of an A.P. is infinite, then it is an infinite A.P.

E.g.: 4, 8,12,16, …

→ If d > 0, then a_{n} > a_{n-1} and if d < 0, then a_{n} < a_{n-1}

→ The general term or nth term of an A.P. is an = a + (n – 1)d.

E.g.: The 10th term of 10, 6, 2, – 2, is Here a = 10; d = a_{2} – a_{1} = 6 -10 = – 4

∴ a_{10} = a + (n – 1)d = 10 + (10 – 1) × – 4 = 10 – 40 + 4 = – 26.

→ Sum of first n – terms of an A.P. is S_{n} = \(\frac{n}{2}\)(a + l), where a is the first term and l is the last term.

E.g : 1 + 2 + 3 + ………… + 80 = \(\frac{80}{2}\)(1 + 80) = 40 × 81 = 3240

→ Sum of the first n – terms of an A.P. is given by,

S_{n} = \(\frac{n}{2}\)[2a + (n – 1)d]. Also, a_{n} = S_{n} – S_{n-1}

→ In a series of numbers, if every number is obtained by multiplying the preceding number by a fixed number except for the first term, such arrangement is called geometric progression or G.P.

E.g.: 4, 8, 16, 32, 64,…

Here, starting from the second term, each term is obtained by multiplying the preceding term by 2. The first term may be denoted by ‘a’, then we also see that

\(\frac{a_2}{a_1}=\frac{a_3}{a_2}=\frac{a_4}{a_3}=\ldots=\frac{a_n}{a_{n-1}}\) = r

We call it “common ratio”, denoted by ‘r’.

→ The general form of a G.P. is a, ar, ar^{2}, ar^{3},…., ar^{n-1}

i.e a_{1} = a, a_{2} = ar, a_{3} = ar^{2}, ……… a_{n} = ar^{n-1}

Important Formula:

- a = First term
- d = T
_{n}– T_{n-1} - T
_{n}= a + (n – 1)d - S
_{n}= \(\frac{n}{2}\)[2a + (n – 1)d] - S
_{n}= \(\frac{n}{2}\)[a + l] - r = \(\frac{T_n}{T_{n-1}}\)
- T
_{n}= ar^{n-1} - S
_{n}= \(\frac{a\left(r^n-1\right)}{r-1}\); r ≥ 1 - S
_{n}= \(\frac{a\left(1-r^n\right)}{1-r}\); r ≤ 1 - S
_{∞}= \(\frac{a}{1-r}\); |r| < 1

Flow Chat:

Carl Fredrich Gauss(1777 – 1855):

- Carl Fredrich Gauss was a German mathematician and physical scientist who contributed significantly to many fields of mathematics.
- Gauss was asked to find the sum of the positive integers from 1 to 100. He immediately replied that the sum is 5050.
- Gauss had a remarkable influence in many fields of mathematics and science and is ranked as one of history’s most influential mathematicians.