Question

Which term of the AP 24 21. 18.15.
Is the first negative term​

Answer 1

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

$\rm :\longmapsto\:Let \: assume \: that \: a_n \: is \: the \: first \: negative \: term.$

Given AP series is

• 24, 21, 18, 15, ----

So,

• First term of an AP, a = 24

• Common difference, d = 21 - 24 = - 3

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.

As,

$\rm :\longmapsto\:a_n < 0$

On substituting the values, we get

$\rm :\longmapsto\:a + (n - 1)d < 0$

$\rm :\longmapsto\:24 + (n - 1)( - 3) < 0$

$\rm :\longmapsto\:24 - 3n + 3 < 0$

$\rm :\longmapsto\:27 - 3n< 0$

$\rm :\longmapsto\:27 < 3n$

$\rm :\longmapsto\:9 < n$

$\bf\implies \:n > 9$

$\bf\implies \:n = 10, \: 11, \: 12, \: - - - -$

Hence,

• 10th term is the first negative term.

Additional Information :-

Sum of n terms of an AP is given by

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

Wʜᴇʀᴇ,

• Sₙ is the sum of n terms.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.

Answer 2

Step-by-step explanation:

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

$\rm :\longmapsto\:Let \: assume \: that \: a_n \: is \: the \: first \: negative \: term.$

Given AP series is

24, 21, 18, 15, ----

So,

First term of an AP, a = 24

Common difference, d = 21 - 24 = - 3

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.

As,

$\rm :\longmapsto\:a_n < 0$

On substituting the values, we get

$\rm :\longmapsto\:a + (n - 1)d < 0$

$\rm :\longmapsto\:24 + (n - 1)( - 3) < 0$

$\rm :\longmapsto\:24 - 3n + 3 < 0$

$\rm :\longmapsto\:27 - 3n< 0$

$\rm :\longmapsto\:27 < 3n$

$\rm :\longmapsto\:9 < n$

$\bf\implies \:n > 9$

$\bf\implies \:n = 10, \: 11, \: 12, \: - - - -$

Hence,

10th term is the first negative term.

Additional Information :-

Sum of n terms of an AP is given by

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

Wʜᴇʀᴇ,

Sₙ is the sum of n terms.

a is the first term of the sequence.

n is the no. of terms.

d is the common difference.