Question

check wheather 205 belongs to Arthematic progression 7,13,19​

Answer 1

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

↝ Given AP series is

↝ 7, 13, 19, -----

It implies,

• First term of an AP, a = 7

• Common difference of an AP, d = 13 - 7 = 6.

Let assume that

• nth term of an AP, aₙ = 205

We know that,

↝ 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.

Thus,

$\rm :\longmapsto\:205 = 7 + (n - 1)6$

$\rm :\longmapsto\:205 - 7 = (n - 1)6$

$\rm :\longmapsto\:198 = (n - 1)6$

$\rm :\longmapsto\:n - 1 = 33$

$\bf\implies \:n = 34$

Hence,

$\: \underbrace{ \boxed{ \sf \: 205 \: is \: {34}^{th} \: term \: of \: AP \: series \: 7, 13, 19, - - - }}$

Additional Information :-

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ Sum of n  terms of an arithmetic sequence is,

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

Wʜᴇʀᴇ,

• Sₙ is the sum of n terms of AP.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.