Question

The 5th term of an A.P is 1/10 and 10th term is 1/5 . Find 50th term of an A.P.​

Answer 1

Answer:

1

Step-by-step explanation:

Let the first term of the AP be a and common difference between the terms be d. [Using nth term = a + (n - 1)d]

5th term = 1/10

=> a + 4d = 1/10 => a = 1/10 - 4d

10th term = 1/5

=> a + 9d = 1/5 => 1/10 - 4d + 9d = 1/5

=> 5d = 1/5 - 1/10 => 5d = 1/10

=> d = 1/50

Hence, a = 1/10 - 4(1/50) = 1/50

Therefore,

50th term = a + 49d = 1/50 + 49(1/50)

50th term = (1 + 49)/50 = 1

Answer 2

$\begin{gathered}\begin{gathered}\bf \:Given-\begin{cases} &\sf{a_5 \: of \: an \: A.P. = \dfrac{1}{10} } \\ \\ &\sf{a_{10} \: of \: an \: A.P. = \dfrac{1}{5} } \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \: To \: Find - \begin{cases} &\sf{a_{50} \: of \: an \: A.P.}\end{cases}\end{gathered}\end{gathered}$

$\large\underline{\sf{Solution-}}$

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ nᵗʰ term of an arithmetic sequence is,

$\begin{gathered}\:\:{\underline{{\boxed{\bf{{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• aₙ is the nᵗʰ term.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.

Tʜᴜs,

↝ 5ᵗʰ term is, given by

$\rm :\longmapsto\:a_5 = a + (5 - 1)d$

$\rm :\longmapsto\:a_5 = a + 4d$

↝ It is given that

$\rm :\longmapsto\:a_5 \: = \: \dfrac{1}{10}$

$\bf\implies \:a + 4d = \dfrac{1}{10} - - - (1)$

Aɢᴀɪɴ,

↝ 10ᵗʰ term is, given by

$\rm :\longmapsto\:a_{10} \: = a + (10 - 1)d$

$\rm :\longmapsto\:a_{10} = a + 9d$

↝ It is given that

$\rm :\longmapsto\:a_{10} = \dfrac{1}{5}$

$\bf\implies \:a + 9d = \dfrac{1}{5} - - - (2)$

↝ On Subtracting equation (1) from equation (2), we get

$\rm :\longmapsto\:a + 9d - a - 4d = \dfrac{1}{5} - \dfrac{1}{10}$

$\rm :\longmapsto\:5d = \dfrac{2 - 1}{10}$

$\rm :\longmapsto\:5d = \dfrac{1}{10}$

$\bf\implies \:d = \dfrac{1}{50} - - - (3)$

↝ On substituting the value of d in equation (1), we get

$\rm :\longmapsto\:a + 4 \times \dfrac{1}{50} = \dfrac{1}{10}$

$\rm :\longmapsto\:a = \dfrac{1}{10} - \dfrac{4}{50}$

$\rm :\longmapsto\:a = \dfrac{5 - 4}{50}$

$\bf\implies \:a = \dfrac{1}{50} - - - (4)$

Now,

↝ 50ᵗʰ term is, given by

$\rm :\longmapsto\:a_{50} = a + (50 - 1)d$

$\rm :\longmapsto\:a_{50} = \dfrac{1}{50} + 49 \times \dfrac{1}{50}$

$\rm :\longmapsto\:a_{50} = \dfrac{1 + 49}{50}$

$\rm :\longmapsto\:a_{50} = \dfrac{50}{50}$

$\bf\implies \:a_{50} \: = \: 1$

Additional Information :-

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ Sum of first n terms of an arithmetic sequence is,

$\underline{\boxed{ \bf \: S_n = \dfrac{n}{2}\bigg( 2a + (n - 1)d\bigg)}}$

Wʜᴇʀᴇ,

• Sₙ is the sum of first 'n' terms.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.