Question

5th term of an arithmetic sequence is 21 and it's 9th term 37. What is the 1st term

Answer 1

EXPLANATION.

5th term of an A.P = 21.

9th term of an A.P = 37.

As we know that,

General term of an A.P.

Tₙ = a + (n - 1)d.

⇒ T₅ = 21.

⇒ a + (5 - 1)d = 21.

⇒ a + 4d = 21 ⇒ (1).

⇒ T₉ = 37.

⇒ a + (9 - 1)d = 37.

⇒ a + 8d = 37 ⇒ (2).

From equation (1) & (2), we get.

⇒ a + 4d = 21.

⇒ a + 8d = 37.

We get,

⇒ - 4d = - 16.

⇒ d = 16/4.

⇒ d = 4.

Put the value of d = 4 in equation (1), we get.

⇒ a + 4d = 21.

⇒ a + 4(4) = 21.

⇒ a + 16 = 21.

⇒ a = 21 - 16.

⇒ a = 5.

First term = a = 5.

Common difference = d = 4.

                                                                                                                           

MORE INFORMATION.

Supposition of terms in A.P.

(1) = Three terms as : a - d, a, a + d.

(2) = Five terms as : a - 2d, a - d, a, a + d, a + 2d.

(3) = Four terms as : a - 3d, a - d, a + d, a + 3d.

Answer 2

$\begin{gathered}\begin{gathered}\bf \:Given - \begin{cases} &\sf{a_5 \: of \: an \: AP = 21} \\ &\sf{a_9 \: of \: an \: AP = 37} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \: To \: find - \begin{cases} &\sf{a_1 \: of \: an \: AP}  \end{cases}\end{gathered}\end{gathered}$

Concept Used :-

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ nᵗʰ term of an arithmetic sequence is,

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

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 21

$\rm :\implies\:a \: + \: (5 - 1)d \: = \: 21$

$\rm :\implies\:a \: + \: 4d = 21$

$\rm :\implies\:a \: = \: 21 \: - \: 4d - - (1)$

Now,

↝ 9ᵗʰ term is 37

$\rm :\implies\:a \: + \: (9 - 1)d = 37$

$\rm :\implies\:a \: + \: 8d \: = 37$

$\rm :\implies\:21 - 4d + 8d \: = \: 37 \: \: \: \: \: { \green{(\because \: a \: = 21 - 4d)}}$

$\rm :\implies\:4d \: = \: 16$

$\boxed{ \pink{\rm :\implies\:d \: = \: 4}}$

On substituting d = 4, in equation (1) is

$\rm :\implies\: \boxed{ \pink{ \bf \: a = 21 - 4 \times 4 \: = 21 - 16 = \tt \: 5}}$