The middle term of arithmatic series 2, 6, 10, ...,146

Concept:

Let us consider sequence a₁, a₂, a₃ …. a_n is an A.P.

• Common difference “d”= a₂ – a₁ = a₃ – a₂ = …. = a_n – a_{n – 1}

• n^th term of the A.P. is given by a_n = a + (n – 1) d

Calculation:

Given series is 2, 6, 10, ...,146

First term, a = 2, last term, a_n = 146, an

Common difference d = 4, so it is an AP

an = a + (n – 1) d

146 = 2 + (n - 1) (4)

⇒ n - 1 = 144/4

⇒ n = 36 + 1 = 37

So, number of terms in given series = 37

Middle term = (37 + 1)/2 = 19th term 

a₁₉ = 2 + (19 - 1) × 4

= 2 + 72 

= 74

Hence, option (3) is correct.