Answer

Verified

456k+ views

**Hint:**Arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.

Difference here means the second minus the first term.

If the initial term of an arithmetic progression is \[{a_1}\]and the common difference of successive members is d them the \[{n^{th}}\] term of the sequence \[{a_n}\] is given by

\[{a_n} = {a_1} + (n - 1)d\]

The behaviour of the AP depends on the common difference d. If the common difference is

\[ \to \]Positive, then the terms will grow towards positive infinity.

\[ \to \] Negative, then the term will grow towards negative infinity.

Now, In the formula

\[{a_n} = {a_1} + (n - 1)d\]

\[ \to {a_1}\] is the first term of an AP

\[ \to \] \[{a_n}\] is the \[{n^{th}}\] term of an AP

\[ \to \] d is the difference between terms of the AP

\[ \to \] n is the number of terms in the AP.

**Complete step-by-step answer:**

Given AP is

\[7,13,19,....205\]

Using the formula

\[{a_n} = a + (n - 1)d\]

Here, first term of AP is

\[a = 7\]

To find the value of $d$ we need to subtract the second term from first term.

\[d = (Second\,term - First\,term)\]

\[d = 13 - 7 = 6\]

Last term is \[an = 205\]

Now, find A.P.

\[{a_n} = a(n - 1)d\]

Put values of each,

We get,

\[\begin{gathered}

205 = 7 + (n - 1) \times 6 \\

205 - 7 = (n - 1) \times 6 \\

198 = (n - 1) \times 6 \\

\end{gathered} \]

Simplify

\[\begin{gathered}

\dfrac{{198}}{6} = n - 1 \\

33 = n - 1 \\

33 + 1 = n \\

34 = n \\

\end{gathered} \]

We have,

$n = 34$

Hence there are 34 terms in the given AP.

**Note:**The sum of the members of finite AP is called on arithmetic series.

For example, consider the sum:

\[2 + 5 + 8 + 11 + 14\]

This sum can be found quickly by taking the number ‘n’ of term bring added (here 5), Multiplying by the sum of the first and last number in the progression and dividing by 2.

\[\begin{gathered}

\dfrac{{n({a_1} + {a_n})}}{2}\,\,\, \\

n = 5 \\

{a_1} = 2 \\

an = 14 \\

\end{gathered} \]

In the case above, this gives the equation

\[2 + 5 + 8 + 11 + 14 = \dfrac{{5(2 + 14)}}{2} = 40\]

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

A rainbow has circular shape because A The earth is class 11 physics CBSE

Which are the Top 10 Largest Countries of the World?

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

How do you graph the function fx 4x class 9 maths CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

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

Why is there a time difference of about 5 hours between class 10 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell