Question

write down the first four terms of the sequence whose general term is t1=2, tn=tn-1+5, n>=2​

Answer 1

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

Given that,

$\rm \: t_1 = 2$

and

$\rm \: t_n = t_{n - 1} + 5 , \: \: \: n \geqslant 2$

So, Substitute n = 2, we get

$\rm \: t_2 = t_{2 - 1} + 5$

$\rm \: t_2 = t_{1} + 5$

$\rm \: t_2 = 2+ 5$

$\rm\implies \:\rm \: t_2 = 7$

Now, Substituting n = 3, we get

$\rm \: t_3 = t_{3 - 1} + 5$

$\rm \: t_3 = t_{2} + 5$

$\rm \: t_3 = 7 + 5$

$\rm\implies \:\rm \: t_3 =12$

Now, On substituting n = 4, we get

$\rm \: t_4 = t_{4 - 1} + 5$

$\rm \: t_4 = t_{3} + 5$

$\rm \: t_4 = 12+ 5$

$\rm\implies \:\rm \: t_4 = 17$

Hence, The first four term of the sequence are

$\rm \: t_1 = 2$

$\rm \: t_2 = 7$

$\rm \: t_3 = 12$

$\rm \: t_4 = 17$

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

Additional Information :-

1. Arithmetic mean between two positive real numbers a and b is given by

$\boxed{\sf{  \:\rm \: AM \: = \: \frac{a + b}{2} \: \: }}$

2. Geometric mean between two positive real numbers a and b is given by

$\boxed{\sf{  \:\rm \: GM \: = \: \sqrt{ab} \: \: }}$

3. Harmonic mean between two positive real numbers a and b is given by

$\boxed{\sf{  \:\rm \: HM = \frac{2ab}{a + b} \: \: }}$

4. Relationship between Arithmetic mean, Geometric mean and Harmonic mean

$\boxed{\sf{  \:\rm \: AM \: \geqslant \: GM \: \geqslant \: HM \: \: }}$

$\boxed{\sf{  \:\rm \: {GM}^{2} \: = \: AM \: \times \: HM \: \: }}$

Answer 2

Step-by-step explanation:

$\color{maroon} \pmb{\[ \begin{aligned} \tt T_{1}-2, T_{n} & \tt=T_{n-1}+5, n \geq 2 \\ \tt \Rightarrow \quad T_{2} & \tt=T_{2-1}+5 \\ & \tt=T_{1}+5=2+5=7 \\ \tt T_{3} & \tt=T_{3-1}+5 \\ & \tt=T_{2}+5=7+5=12 \\ \tt \text { and } T_{4} & \tt=T_{4-1}+5 \\ & \tt=T_{3}+5=12+5=17 \end{aligned} \] }$

$\color{blue}{ \text{\( \therefore \) Ist four terms are 2, 7,12 and \( \tt 17 . \)}}$