Question

$\sf \: If \: 73 \: is \: the {n}^{t \: h} \: term \: of \: the \: ap \: 3,8,13,18 \: then \: n \: is$
​

Answer 1

Arithmetic Progression

We will use the concept of AP to solve the required problem.

AP - A sequence of numbers in which common difference between two consecutive terms is always same.

Step-by-step solution:

The given arithmetic progression is $3,8,13,18$ and the nth term of the AP is $73$.

Here, the first term $a_1 = 3$ and the second term $a_2 = 8$.

We know that, the formula for finding common difference is:

$\boxed{d = a_2 - a_1}$

By substituting the known values in the formula, we get:

$\implies d = 8 - 3$

$\implies d = 5$

Now, we know that, In an AP with first term $a$ and common diffence $d$, then $nth$ term is given by,

$\boxed{a_n = a + (n - 1) d}$

By substituting the known values in the formula, we get the following results:

$\implies 73 = 3 + (n - 1) \times 5$

$\implies 73 = 3 + (n - 1) \times 5$

$\implies 73 = 3 + 5n - 5$

$\implies 73 = 5n - 2$

$\implies 73 + 2 = 5n$

$\implies 75 = 5n$

$\implies \boxed{n = 15}$

Hence, the value of n term is 15.

$\rule{300}{2}$

Extra Information:

1. In an AP with first term $a$ and common diffence $d$, then $nth$ term is given by,

$\longrightarrow \boxed{T_n = a + (n - 1) d}$

2. Let $a$ be the first term, $d$ be the common difference and $l$ be the last term if an AP. Then nth term from the end is given by,

$\longrightarrow \boxed{\rm{nth \; term \; from \; the \; end} = \it{l - (n - 1) d}}$

3. The sum of $n$ terms of an AP in which first term $a$, common diffence $d$ and last term $l$ is given by,

$\longrightarrow \boxed{S_n = \dfrac{n}{2} (a + l)}$

Answer 2

Answer:

Given :-

• The nth term of an AP is 3, 8, 13, 18 is 73.

To Find :-

• What is the value of n or number of terms of an AP.

Formula Used :-

$\clubsuit$ General term or nth term of an AP Formula :

$\bigstar \: \: \sf\boxed{\bold{\pink{a_n =\: a + (n - 1)d}}}\: \: \: \bigstar$

where,

• $\sf a_n$ = nth term of an AP
• a = First term of an AP
• n = Number of terms of an AP
• d = Common Difference of an AP

Solution :-

First, we have to find the common difference (d) :-

Given :

• a₁ = 3
• a₂ = 8

According to the question :

$\implies \bf Common\: Difference(d) =\: a_2 - a_1$

$\implies \sf Common\: Difference(d) =\: 8 - 3$

$\implies \sf\bold{\purple{Common\: Difference(d) =\: 5}}$

Hence, the common difference or d is 5 .

Now, we have to find the value of n :

Given :

• nth term $\sf (a_n)$ = 73
• First term (a) = 3
• Common Difference (d) = 5

According to the question by using the formula we get,

$\dashrightarrow \bf a_n =\: a + (n - 1)d$

$\dashrightarrow \sf 73 =\: 3 + (n - 1)5$

$\dashrightarrow \sf 73 - 3 =\: (n - 1)5$

$\dashrightarrow \sf 70 =\: (n - 1)5$

$\dashrightarrow \sf \dfrac{\cancel{70}}{\cancel{5}} =\: (n - 1)$

$\dashrightarrow \sf \dfrac{14}{1} =\: (n - 1)$

$\dashrightarrow \sf 14 =\: n - 1$

$\dashrightarrow \sf - n =\: - 1 - 14$

$\dashrightarrow \sf {\cancel{-}} n =\: {\cancel{-}} 15$

$\dashrightarrow \sf\bold{\red{n =\: 15}}$

$\therefore$ The value of n or number of terms of an AP is 15 .

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EXTRA IMPORTANT FORMULA :-

$\diamond$ General term or nth term of an AP Formula :

$\bigstar \: \: \sf\boxed{\bold{\pink{a_n =\: a + (n - 1)d}}}\: \: \: \bigstar$

where,

• $\sf a_n$ = nth term of an AP
• a = First term of an AP
• n = Number of terms of an AP
• d = Common Difference of an AP

$\diamond$ Sum of nth term of an AP Formula :

$\bigstar \: \: \sf\boxed{\bold{\pink{S_n =\: \dfrac{n}{2}\bigg\lgroup 2a + (n - 1)d\bigg\rgroup }}}\: \: \: \bigstar$

where,

• $\sf S_n$ = Sum of nth term of an AP
• n = Number of terms of an AP
• a = First term of an AP
• d = Common Difference of an AP