Question

write an a.p. whose first term is a and common difference is d in each of the following
1) a = –7,d = one upon two​

Answer 1

$\begin{gathered}\begin{gathered}\bf \:Given -\begin{cases} &\sf{An \: A.P. \: series \: in \: which } \\ &\sf{first \: term \: (a) = - \: 7}\\ &\sf{common \: difference \: (d) \: = \: \dfrac{1}{2} } \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \:To\: find - \begin{cases} &\sf{An \: A.P. \: series}  \end{cases}\end{gathered}\end{gathered}$

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

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,

According to statement,

It is given that

$\tt \: a_1 \: = \: - \: 7 \: \: and \: \: d \: = \: \dfrac{1}{2}$

So,

Second term, is

$\tt \: a_2 \: = \: a \: + \: (2 - 1) \: d$

$\tt \: a_2 \: = \: a \: + \: d$

$\bf \: a_2 \: = \: - \: 7 \: + \: \dfrac{1}{2} = \dfrac{ - 14 + 1}{2} = - \: \dfrac{13}{2}$

Third term, is

$\tt \: a_3 \: = \: a \: + \: (3 \: - \: 1) \: d$

$\tt \: a_3 \: = \: a \: + \: 2 \: d$

$\bf \: a_3 \: = \: - \: 7 \: + \cancel2 \times \dfrac{1}{\cancel2} = - 7 + 1 = - \: 6$

Fourth term, is

$\tt \: a_4 \: = \: a \: + \: (4 \: - \: 1) \: d$

$\tt \: a_4 \: = \: a \: + \: 3 \: d$

$\bf \: a_4 \: = \: - \: 7 \: + \: 3 \times \dfrac{1}{2} = \dfrac{ - 14 + 3}{2} = - \: \dfrac{11}{2}$

Hence,

The required A.P. series is

$\bf \: - 7, \: - \: \dfrac{13}{2} , \: - 6, \: - \: \dfrac{11}{2} , \: - - - -$

Additional Information :-

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

$\begin{gathered}\:\:{\underline{{\boxed{\bf{{S_n\:=\dfrac{n}{2}( \:a\:+\:a_n)}}}}}} \\ \end{gathered}$

$\bf \: 3 \: numbers \: in \: A.P. \: are \: a - d,a,a + d$

$\bf \: 4 \: numbers \: in \: A.P. \: are \: a - 3d,a - d,a + d,a + 3d$