TS 10th Class Maths Notes Chapter 6 Progressions

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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 = a₂ – a₁ = a₃ – a₂ = a₄ – a₃ = ………………… = 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 = a₂ – a₁ = 6 -10 = – 4
∴ a₁₀ = 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, ar², ar³,…., ar^n-1
i.e a₁ = a, a₂ = ar, a₃ = ar², ……… 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.