\( ) \) \( a n = \)

Let $a_n=5^n+7^n$, and
$\Delta =\left|\begin{array}{ccc}{a}_{n+1}& a_{n+2}& a_{n+3}\\ a_{n+4}& a_{n+5}& a_{n+6}\\ a_{n+7}& a_{n+8}& a_{n+9}\end{array}\right|$
then $\Delta$ equals

If $a_1,a_2\dots ,a_{2n+1}$ are in A.P., then $\frac{a_{n+2}-a_n}{a_{n+2}+a_n}+\frac{a_{n+3}-a_{n-1}}{a_{n+3}+a_{n-1}}+\cdots +\frac{a_{2n+1}-a_1}{a_{2n+1}+a_1}$ is equal to

is \varnothing is latitude and ∂ is dip at a place then 1. †an\varnothing=†an∂/2 2. †an ∂= †an\varnothing/2 3.†an ∂= 1/†an\varnothing 4. †an^2\varnothing+ †an^2∂=

A sequence a₁, a₂, a₃,…., a_n,…. is called arithmetic progression, if _______ where a₁ is called the first term and the constant term d is called the common difference of the A.P.

Write the first five terms of each of the following sequences whose nth terms are:

(a) a_n = 3n + 2

(b) $a_n=\frac{n-3}{3}$

(c) a_n = 3ⁿ

(d) $a_n=\frac{3n-2}{5}$

(e) a_n = (−1)ⁿ 2ⁿ

(f) $a_n=\frac{n(n-2)}{2}$

(g) an =n² − n + 1

(h) a_n = 2n² − 3n + 1

(i) $a_n=\frac{2n-3}{6}$