Question

find the number of terms in the AP. 18,15¹,13,....,-47​

Answer 1

Step-by-step explanation:

Since we have given that

,13,......−47

18,

2

31

,13,.........−47

Here, first term = a = 15

d = common difference is given by

\begin{gathered}a_2-a_1\\\\=\frac{31}{2}-18\\\\=\frac{31-36}{2}=\frac{-5}{2}=-2.5\end{gathered}

a

2

−a

1

=

2

31

−18

=

2

31−36

=

2

−5

=−2.5

And the last term is given by

a_n=-47a

n

=−47

We need to find the number of terms 'n':

\begin{gathered}a_n=a+(n-1)d\\\\-47=18+(n-1)(-2.5)\\\\-47-18=(n-1)(-2.5)\\\\-65=(n-1)(-2.5)\\\\\frac{65}{2.5}=n-1\\\\26=n-1\\\\n=27\end{gathered}

a

n

=a+(n−1)d

−47=18+(n−1)(−2.5)

−47−18=(n−1)(−2.5)

−65=(n−1)(−2.5)

2.5

65

=n−1

26=n−1

n=27

Hence, there are 27 terms in the sequence.

Answer 2

Question :

Correct question : find the number of terms in the AP. 18,15¹/₂,13,....,-47​

Answer :

number of terms = 27

Step-by-step explanation :

   $\underline{\underline{\bf Arithmetic \ Progression(AP) :}}$

•    It is the sequence of numbers such that the difference between any two successive numbers is constant.

•    In AP,

           a - first term

           d - common difference

           aₙ - nth term

           Sₙ - sum of n terms

•   General form of AP,

                 a , a+d , a+2d , a+3d , ..........

•   Formulae :-

             $\bigstar$  nth term of AP,

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

             $\bigstar$  Sum of n terms,

                 $\boxed{\bf S_n=\frac{n}{2}[2a+(n-1)d]}$

__________________________________

Coming back to the question,

               

                Given AP series,

                18 , 15¹/₂ , 13 ,...., -47

✯ first term, a = 18

✯ common difference, d = 15¹/₂ - 18 = 13 - 15¹/₂ = -2¹/₂ = -5/2

✯ last term, l = -47

     we have to find the number of terms in the given AP.

     Let nth term be "-47" (since it is the last term)

nth term, $\boxed{\bf a_n=a+(n-1)d}$

                $\bf -47=18+(n-1)(\frac{-5}{2}) \\\\ -47-18=\frac{-5n+5}{2} \\\\-65=\frac{-5n+5}{2}\\\\ -65 \times 2=-5n+5\\\\-130=-5n+5\\\\ -130-5=-5n\\\\ -135=-5n \\\\ 5n=135\\\\n=\frac{135}{5}\\\\n=27$

∴ There are 27 terms in the AP : 18 , 15¹/₂ , 13 , ...... , -47