Answer

Verified

420.9k+ views

**Hint:**Here we will simply apply the formula of the sum of the $n$ terms of $AP$ which is the Arithmetic progression and get the value of the common difference by this formula. Once we get the common difference $d$ then we can simply solve for $20{\text{th}}$ term by applying the formula of the $n{\text{th}}$ term of $AP$

**Formula Used:**

${\text{sum of n terms}}({S_n}) = \dfrac{n}{2}(2a + (n - 1)d)$

$n{\text{th term}} = {a_n} = a + (n - 1)d$

**Complete step-by-step answer:**AP or arithmetic progression is the sequence in which the different terms have the same common difference or we can say that the consecutive numbers differ by the same number. For example: in the sequence like $2,4,6,8,.......100$ we can see that the difference between each consecutive term is $2$ as $4 - 2 = 6 - 4 = 8 - 6 = 2$

Hence the given sequence is called the Arithmetic progression or AP

Here we are given that the sum of first $14$ terms of$AP$ is $1050$

So we can apply the formula of the sum of the $n$ terms which is

${\text{sum of n terms}}({S_n}) = \dfrac{n}{2}(2a + (n - 1)d)$

We know that

${\text{Sum}} = 1050$

$a = {\text{first term}} = 10$

$n = {\text{number of terms}} = 14$

Here $d = $common difference

So substituting the values in the formula we get:

${\text{sum of n terms}}({S_n}) = \dfrac{n}{2}(2a + (n - 1)d)$

$\Rightarrow$ $1050 = \dfrac{{14}}{2}(2(10) + (14 - 1)d)$

$\Rightarrow$ $1050 = 7(20 + 13d)$

$\Rightarrow$ $\dfrac{{1050}}{7} = 20 + 13d$

$\Rightarrow$ $150 = 20 + 13d$

$\Rightarrow$ $13d = 130$

$\Rightarrow$ $d = 10$

Hence we get that the common difference of the given arithmetic progression is $10$ which means that each term of the given sequence is $10$ more than the previous one.

Now we have got the common difference and now we need to know the $20{\text{th}}$ term of the sequence.

So we apply the formula of the $n{\text{th}}$ term of the AP we get

$n{\text{th term}} = {a_n} = a + (n - 1)d$

Here we should know what the value of each variable in the question is:

${a_n} = n{\text{th term}}$

And $a = $first term$ = 10$

And $n = 20$ as we need to find the $20{\text{th}}$ term

And $d = 10$ as we had calculated earlier

${a_n} = 10 + (20 - 1)10$

$

= 10 + (19)(10) \\

= 10 + 190 \\

= 200 \\

$

**Hence we get that the $20{\text{th}}$ term of the sequence which is in AP is $200$.**

**Note:**Here we need to understand the meaning of the Arithmetic progression and we should know what formula should be used in order to calculate the $n{\text{th}}$ term and the sum of the n terms. We should not make calculation mistakes as these types of questions are simple but need just the formula and the values of the parameters used in the formula.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Which are the Top 10 Largest Countries of the World?

Difference Between Plant Cell and Animal Cell

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE