Question

Among the following an A.P is

A)2,3,5,7 and so on
B)2,5,7,10 and so on
C)-1,-3,-5,-7 and so on
D)both B,C ​

Answer 1

Answer:

AP: the difference between every successive(next term) and the preceding(previous term) should be equal.

Here in option A:  3-2 = 1 and 5-3 = 2 and 7-5 = 2 so it is not an Ap because we got 1, 2,2

option b: 5-2 = 3, 7-5 = 2, 10-7 = 3, so it is not an AP because we got 3, 2, 3

option c: -3 -(-1) = -2, -5-(-3) = -2 and -7-(-5) = -2

In option c we got -2, -2,-2 when we tried to subtract every successive term with the previous term so option c is in AP

Step-by-step explanation:

Answer 2

$\large\underline{\sf{Solution-}}$

Before we proceeding with the question, Let's recall whats an AP series.

The sequence of terms in which the difference between the two consecutive terms remains constant through out is called an Arithmetic Progression or AP.

Now, we have to justify the given statement for this.

Consider, $\red{\rm\:2, 3, 5, 7, \cdots\cdots}$

$\rm \: a_2 - a_1 = 3 - 2 = 1$

$\rm \: a_3 - a_2 = 5 - 3 = 2$

$\rm\implies \:a_3 - a_2 \: \ne \: a_2 - a_1$

$\bf\implies \:2,3,5,7, \cdots \cdots \: is \: not \: an \: AP$

Now, Consider $\red{\rm\:2, 5, 7, 10, \cdots\cdots}$

$\rm \: a_2 - a_1 = 5 - 2 = 3$

$\rm \: a_3 - a_2 = 7 - 5 = 2$

$\rm\implies \:a_3 - a_2 \: \ne \: a_2 - a_1$

$\bf\implies \:2,5,7,10, \cdots \cdots \: is \: not \: an \: AP$

Consider, $\red{\rm\:-1, -3, -5, -7, \cdots\cdots}$

$\rm \: a_2 - a_1 = - 1 - ( - 3) = - 1 + 3 = 2$

$\rm \: a_3 - a_2 = - 3 - ( - 5) = - 3 + 5 = 2$

$\rm \: a_4 - a_3 = - 5 - ( - 7) = - 5 + 7 = 2$

$\rm\implies \:a_3 - a_2 \: = \: a_2 - a_1$

$\bf\implies \: - 1, - 3, - 5, - 7, \cdots \cdots \: is \: an \: AP$

$\rule{190pt}{2pt}$

$\red{\rm\:ADDITIONAL\:INFORMATION}$

↝ nᵗʰ term of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• aₙ is the nᵗʰ term.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.