Question

Progressions and series: 30 points. Please help. I need proper explanation.
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Thanks a lot.​

Answer 1

Hello dear.

What's Your que from progression ??

...

Answer 2

Step-by-step explanation:

Progressions (AP, GP, HP)

• Progressions (or Sequences and Series) are numbers arranged in a particular order such that they form a predictable order. By predictable order, we mean that given some numbers, we can find next numbers in the series.
• Arithmetic Progression (AP)
• A sequence of numbers is called an arithmetic progression if the difference between any two consecutive terms is always same. In simple terms, it means that next number in the series is calculated by adding a fixed number to the previous number in the series. This fixed number is called the common difference.
• For example, 2,4,6,8,10 is an AP because difference between any two consecutive terms in the series (common difference) is same (4 – 2 = 6 – 4 = 8 – 6 = 10 – 8 = 2).
• If ‘a’ is the first term and ‘d’ is the common difference,
• nth term of an AP = a + (n-1) d
• Arithmetic Mean = Sum of all terms in the AP / Number of terms in the AP
• Sum of ‘n’ terms of an AP = 0.5 n (first term + last term) = 0.5 n [ 2a + (n-1) d ]
• Geometric Progression (GP)
• A sequence of numbers is called a geometric progression if the ratio of any two consecutive terms is always same. In simple terms, it means that next number in the series is calculated by multiplying a fixed number to the previous number in the series. This fixed number is called the common ratio.
• For example, 2,4,8,16 is a GP because ratio of any two consecutive terms in the series (common difference) is same (4 / 2 = 8 / 4 = 16 / 8 = 2).
• If ‘a’ is the first term and ‘r’ is the common ratio,
• nth term of a GP = a rn-1
• Geometric Mean = nth root of product of n terms in the GP
• Sum of ‘n’ terms of a GP (r < 1) = [a (1 – rn)] / [1 – r]
• Sum of ‘n’ terms of a GP (r > 1) = [a (rn – 1)] / [r – 1]
• Sum of infinite terms of a GP (r < 1) = (a) / (1 – r)
• Harmonic Progression (HP)
• A sequence of numbers is called a harmonic progression if the reciprocal of the terms are in AP. In simple terms, a,b,c,d,e,f are in HP if 1/a, 1/b, 1/c, 1/d, 1/e, 1/f are in AP.
• For two terms ‘a’ and ‘b’,
• Harmonic Mean = (2 a b) / (a + b)
• For two numbers, if A, G and H are respectively the arithmetic, geometric and harmonic means, then
• A ≥ G ≥ H
• A H = G2, i.e., A, G, H are in GP