The sum of three numbers in a GP is 42 . If the first two numbers are increased by 2 and the thi

The sum of three numbers in a GP is 42 . If the first two...

Key Concepts

Algebraic Equation Setup

Setting up algebraic equations based on the properties of sequences is crucial in solving such problems. By representing terms using variables and known relationships (like sums or differences), one can form equations that capture all conditions of the problem and solve for the unknowns.

Sequence Transformation

Sequence transformation refers to the process of applying operations such as addition or subtraction to the terms of a sequence, which can change its defining properties. Recognizing how changes to specific terms affect the overall pattern is key to establishing new relationships like converting a geometric progression to an arithmetic progression.

Arithmetic Progression

An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant, known as the common difference. This concept is useful when determining relationships between altered terms of a sequence after applying specific modifications.

Geometric Progression

A geometric progression is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a constant called the common ratio. Understanding this concept allows one to represent the sequence in a general form and apply the sum condition effectively.

Transcript

00:03 Hello, so the question is taken from sequence and series given that the sum of the three numbers in a gp is 42 so given that's let three numbers in gps are a a r and a r square where r is the common ratio so given that a plus a r plus a square is equal to 42 this implies a is equal to 42 by one plus r plus r plus square okay so and the next condition is given that if the first two numbers are increased by two and the third is decreased by four the resulting number forms and api find the number so given that if the first two number it means a is increased by two ar is also increased by two and a r square is decreased by four then they form an ap condition of ap is a b minus a is equal to c minus b ar minus a is equal to a r square minus a r minus two minus four so that will be if we take all of a to the left hand side we get a r take this to the right hand side and six to the that is and then uh changing the side of both of these tom so a r square minus 2 a r plus a is equal to 6 okay so taking a common we get r square minus 2 r plus 1 is equal to 6 substituting the value of a here and the value of a is 42 by r square plus r plus 1 into r square minus 2 r plus 1 is equal to 6 7 the 42 so we get 7 r square minus 14 r plus 7 is equal to r square plus r plus 1 taking these to the left hand side we get 6 r square minus 15 r plus 6 is equal to 0 taking 3 common we get 2 r 2 r 2 minus 5 r plus 2 is equal to 0 so taking if we valued the value of r r is equal to 2 and r is equal to 1 by 2 so the value of a we get two values of a first one is 42 if r is equal to 2 42 by r square plus r plus 1 so that will be 4 plus 2 plus 1 6 and second value of is 42 by 1 by 4 plus 1 by 2 plus 1 so that will be 42 into 6 take else and we get 6 by 7 we get each value is equal to 1 second not 4 it will be let me evaluate the lcm 4 1 plus 2 plus 7 so 4 by 7 so that will be 4 by 7 okay so 7, 6th of 42 we get 24.

03:39 So from here we get the series.

03:42 First one corresponding to 6.

03:45 6 into 2, 6 into 2 square.

03:50 So that will be 2, 12, 24...

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