Question

Find the sum of AP series whose first term is x, second term is y and last term is z.​

Answer 1

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

Given that,

First term of an AP series = x

Second term of an AP series = y

Last term of an AP series = z

Let assume that a and d represents the first term and common difference of an AP series respectively.

Let number of terms = n

As it is given that,

First term, a = x

Second term = y

So, common difference, d = y - x

Last term, nth term = z

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ nᵗʰ term of an arithmetic progression 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, on substituting the values, we get

$\rm \: z = x + (n - 1)(y - x)$

$\rm \: z - x = (n - 1)(y - x)$

$\rm \: n - 1 = \dfrac{z - x}{y - x}$

$\rm \: n = \dfrac{z - x}{y - x} + 1$

$\rm \: n = \dfrac{z - x + y - x}{y - x}$

$\rm \: n = \dfrac{z + y - 2x}{y - x}$

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ Sum of n  terms of an arithmetic progression is,

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

Wʜᴇʀᴇ,

Sₙ is the sum of n terms of AP.

a is the first term of the sequence.

n is the no. of terms.

So, on substituting the values, we get

$\rm \: S_n = \dfrac{z + y - 2x}{2(y - x)} \: (x + z)$

Hence,

$\rm\implies \:\boxed{\sf{  \:\rm \: S_n = \dfrac{z + y - 2x}{2(y - x)} \: (x + z) \: \: }}$

$\rule{190pt}{2pt}$

Additional Information

↝ Sum of n  terms of an arithmetic progression 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 progression.

n is the no. of terms.

d is the common difference.

Answer 2

Answer:

Given data:-

The first term of the AP series is x.

The second term is y.

The last term is z.

Assume, the number of terms in the AP series in n.

Then, the formula for the last terms of this AP series is given by,

z = x+(n-1) (y-x)

»» z-x = (n-1) (y-x)

»» (z-x)/(y-x) = (n-1)

»» n = (z-x)/(y-x) + 1

»» n = z-x + y-x / y-x

»» n = z+y-2x / y-x

Then, the formula for the sum of the AP series is given by,

S = n/2(x+z)

So,

S = z+y-2x / 2(y-x) (x+z)