Question

the sum of the first n terms of an arithmetic terms of an arithmetic sequence is Sn=bn-2n^2. find b if the tenth term is -16​

Answer 1

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

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.

Thus,

According to statement,

$\rm :\longmapsto\:S_n = bn - {2n}^{2}$

$\rm :\longmapsto\:\dfrac{n}{2} \bigg(2 \:a\:+\:(n\:-\:1)\:d \bigg) = n(b - 2n)$

$\rm :\longmapsto\:2a + nd - d = 2b - 4n$

$\rm :\longmapsto\:(2a - d) + nd = 2b - 4n$

On comparing, we get

$\rm :\longmapsto\:d = - \: 4$

and

$\rm :\longmapsto\:2a - d = 2b$

$\rm :\longmapsto\:2a + 4 = 2b$

$\rm :\longmapsto\:a + 2 = b$

$\rm :\longmapsto\:a= b - 2$

Now,

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,

According to statement,

$\rm :\longmapsto\:a_{10} = - \: 16$

$\rm :\longmapsto\:a + (10 - 1)d= - \: 16$

$\rm :\longmapsto\:a + 9d= - \: 16$

On substituting the values of a and d, we get

$\rm :\longmapsto\:b - 2 + 9( - 4)= - \: 16$

$\rm :\longmapsto\:b - 2 - 36= - \: 16$

$\rm :\longmapsto\:b - 38= - \: 16$

$\rm :\longmapsto\:b = - \: 16 + 38$

$\bf\implies \:b = 22$