a1, a2, a3, …are in A.P. with common difference not a multiple of 3 . Then, maximum number of co

A1, a2, a3, …are in A.P. with common difference not a...

Key Concepts

Arithmetic Progression

An arithmetic progression is a sequence where each term is obtained by adding a constant value, known as the common difference, to the preceding term. This structure underpins the analysis of patterns and properties in sequences, such as the distribution or occurrence of special numbers like primes.

Prime Numbers

Prime numbers are natural numbers greater than 1 that have no divisors other than 1 and themselves. Their unique divisibility properties are central when examining sequences, as the appearance of a composite number can interrupt sequences consisting entirely of primes.

Modular Arithmetic

Modular arithmetic involves calculations with remainders and is instrumental in understanding the periodic behavior of sequences in relation to divisibility. By analyzing the residues of sequence terms modulo a small number, such as 3, one can predict inevitable repetitions that may force composite values.

Divisibility Constraints in Sequences

In any arithmetic progression, the inherent divisibility properties, dictated by the common difference and the starting term, may lead to systematic patterns where some terms must be divisible by certain primes. This concept illustrates why, under specific conditions, an arithmetic progression can only sustain a limited number of consecutive prime numbers before a composite term inevitably appears.

Transcript

00:01 So, in the question, a1, a2 and a3 are in arithmetic progression with a common difference, not a multiple of 3.

00:11 Then the maximum number of consecutive terms so that all the terms are in are prime numbers.

00:20 So, here in the question it is given that in an arithmetic sequence, the common difference, d, which is not equal to multiple of 3.

00:33 Not equal to multiple of three and we can we know that a common difference in an arithmetic a common difference in an arithmetic series is equal to second term minus first term or which is equal to third term minus second term and so on so the series is so the series is first term then first term term plus common difference then first term plus 2 into d and so on here in the question it is it's already said that common difference is not equal to multiple of three that is common difference is equal to three lambda that is or multiple of 3 we can either say that 3m multiple of 3 plus 1 this is not a multiple of 3m plus 1 is not a multiple of 3 so either d is equal to 3m plus 1 or d is equal to 3m plus 2 this also not a multiple of 3 but we can't say 3 is equal to 3m plus 3 is wrong because 3m a multiple of 3 plus 3 is also a multiple of 3 so this is not a chance for common difference so there are two type of common difference are there either 3m plus 1 or 3m plus 2 so now we can put this into the equation that is series is equal to series to a 1 plus d sorry a 1 is first term second term a 1 plus 3m plus 1 3m plus 1 next term a 1 plus 6m plus 2 then a 1 plus 9 of plus 3 and soar or if using the second second term as d then we will get a1 comma a1 plus 3m plus 2 then a 1 plus 6m plus 4 then a 1 plus 9m plus 6 etc etc so these are the terms to check the to check this is a prime number or not we we put a is equal to 1 and m is equal to 1 so when we put m is equal to 1 we will get d is equal to 3m plus 1 which will give 3 into 1 plus 1 which is equal to 4.

04:56 Similarly, apply this value of d into the first equation.

05:03 So put this value of d into the fastication.

05:07 So we will get values like 1, 5, 7, 9, 13, and so on as first series...

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