Statement 1 If a1, a2, a3 are in AP and b1, b2, b3 are in AP then (a1+2 b1),(a2+2 b2),(a3+2 b3

step 1 Hi, so let's work on statement two first. The second two is if A1 plus A2 plus A3 are in AP, the

Statement 1 If a1, a2, a3 are in AP and b1, b2, b3 are in...

Statement 1 If $\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}$ are in $\mathrm{AP}$ and $\mathrm{b}_{1}, \mathrm{~b}_{2}, \mathrm{~b}_{3}$ are in $\mathrm{AP}$ then $\left(a_{1}+2 b_{1}\right),\left(a_{2}+2 b_{2}\right),\left(a_{3}+2 b_{3}\right)$ are in AP and Statement 2 If $a_{1}+a_{2}+a_{3}+\ldots$ are in AP then $\left(a_{1}+k\right)+\left(a_{2}+k\right)+\left(a_{3}+k\right)+\ldots$ are also in $A P$

Statement 1 If $\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}$ are in $\mathrm{AP}$ and $\mathrm{b}_{1}, \mathrm{~b}_{2}, \mathrm{~b}_{3}$ are in $\mathrm{AP}$
then $\left(a_{1}+2 b_{1}\right),\left(a_{2}+2 b_{2}\right),\left(a_{3}+2 b_{3}\right)$ are in AP
and
Statement 2 If $a_{1}+a_{2}+a_{3}+\ldots$ are in AP then $\left(a_{1}+k\right)+\left(a_{2}+k\right)+\left(a_{3}+k\right)+\ldots$ are also in $A P$

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Key Concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference, known as the common difference, defines the linear structure of the sequence, and every term can be expressed based on the first term and the common difference. This concept is central to understanding how modifications to the sequence affect its linearity.

Linear Combination of Arithmetic Progressions

When dealing with arithmetic progressions, applying a linear combination (such as multiplying one sequence by a constant and then adding it to another sequence) results in another arithmetic progression. This is because the operations preserve the constant difference between consecutive terms; scaling and adding fixed multiples to each term of APs maintains the uniform spacing, which is a key property of arithmetic sequences.

Translation Invariance in Sequences

Adding a constant (translating) to each term of an arithmetic progression does not change the common difference between consecutive terms. This invariance means that translating all the terms by the same fixed amount results in an arithmetic progression with the same structure and same common difference, even though each term is shifted by the constant.

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