Question

If 73 is the nth term of the arithmetic progression 3, 8, 13, 18, then ‘n’ is __ ?

Answer 1

Answer:

15

Step-by-step explanation:

Given :

nth term of A.P. 3, 8, 13, 18,... is 73

To find :

The value of 'n'

Solution :

We know that,

$\boxed{\boxed{\begin{array} {cc} \mathbf{T_n = a + (n-1)d}\\\text {where}\\\mathrm{T_n = nth\ term}\\\mathrm{a = first\ term}\\\mathrm{n = number\ of\ terms}\\\mathrm{d = common\ diferrence}\end{array}}}$

So, here, as AP is 3, 8, 13, 18,...

We know that, common difference is the difference of the consecutive terms.

d = 8 - 3 = 13 - 8 = 5

∴ d = 5

Now, a = 3, Tₙ = 73, d = 5

Applying the given formula,

$\implies \mathrm{T_n = a + (n-1)d}$

$\implies \mathrm{73 = 3 + (n-1)(5)}$

$\implies \mathrm{73 - 3 = (n-1)(5)}$

$\implies \mathrm{70 = (n-1)(5)}$

$\implies \mathrm{(n-1) = \dfrac{\cancel{70}^{14}}{\cancel{5}}}$

$\implies \mathrm{n - 1 = 14}$

$\implies \mathrm{n = 14 +1}$

$\therefore \underline{\boxed{\mathbf{n = 15}}}$

❖ Extra information :

⟡ Sum of n terms of an AP is :

$\boxed{\boxed{\begin{array} {cc} \mathbf{S_n = \dfrac{n}{2}\Big(2a + (n-1)d\Big)}\\\text {where}\\\mathrm{S_n = Sum\ of\ n\ terms}\\\mathrm{a = first\ term}\\\mathrm{n = number\ of\ terms}\\\mathrm{d = common\ diferrence}\end{array}}}$

Hope it helps!!

Answer 2

Hint:

If $a_{1}$ be the first term of an Arithmetic Progression and $d$ be its common difference, then

$\quad a_{n}=a_{1}+(n-1)d$

Step-by-step explanation:

Step 1. Determining the first term

Here the progression is $3,8,13,18,...\: ...$

Then the first term, $a_{1}=3$

Step 2. Finding the common difference

Here the first term, $a_{1}=3$ and the second term, $a_{2}=8$

Then the common difference, $d$

$=a_{2}-a_{1}$

$=8-3$

$=5$

Step 3. Finding the n-th term

Since $a_{1}=3$ and $d=5$, we obtain the nth term,

$\quad a_{n}=a_{1}+(n-1)d$

$=3+(n-1)\times 5$

$=3+5n-5$

$=5n-2$

Step 4. Using the given condition

Given that, $73$ is the nth term of the given progression

$\quad 5n-2=73$

$\Rightarrow 5n=73+2$

$\Rightarrow 5n=75$

$\Rightarrow n=15$

Final answer: n = 15

73 is the 15th term of the given progression.

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