If a_1, a_2, a_3, ... and b_1, b_2, b_3, ...



Answered question


If a1,a2,a3,andb1,b2,b3, are arithmetic sequences, show that a1+b1,a2+b2,a3+b3, is also an arithmetic sequence.

Answer & Explanation



Beginner2022-01-05Added 43 answers

Given that the sequences a1,a2,a3, and b1,b2,b3, are arithmetic sequences.
Let the sequence a1,a2,a3, be an arithmetic sequence with common difference d1 (the difference of two consecutive terms of thesequence).
The common difference d1 of the sequence a1,a2,a3, can be calculated as d1=a2a1 or d1=a3a2
Let the sequence b1,b2,b3, be an arithmetic sequence with common difference d2. It can be calculated as d2=b2b1 or d2=b3b2
Consider the sequence a1+b1,a2+b2,a3+b3,
Let d be the common difference of this sequence. It can be calculated as d=(a2+b2)(a1+b1)=(a2a2)+(b2b1)=d1+d2.
Therefore the common difference of the sequence a1+b1,a2+b2,a3+b3, can be written as d=d1+d2
Since, the expression a1+b1,a2+b2,a3+b3, has same common difference between each two consecutive terms. Hence, this expression is an arithmetic sequence if the sequence a1,a2,a3, and the sequence b1,b2,b3, are arithmetic sequences.

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