Question

Find the value of s28– 2s27+ s26​

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

Let in an arithmetic progression

First term = a

Common Difference = d

Then sum of first n terms

$\displaystyle \: S_n = \frac{n}{2} \bigg[ 2a + (n - 1)d \bigg]$

TO DETERMINE

$S_{28} - 2 \times S_{27} + S_{26}$

EVALUATION

Let in an arithmetic progression

First term = a

Common Difference = d

$\displaystyle \: S_{26} = \frac{26}{2} \bigg[ 2a + (26 - 1)d \bigg]$

$\implies \: \displaystyle \: S_{26} = 13\bigg[ 2a + 25d \bigg] = 26a + 325d$

$\displaystyle \: S_{27} = \frac{27}{2} \bigg[ 2a + (27 - 1)d \bigg]$

$\implies \: \displaystyle \: S_{27} = \frac{27}{2} \bigg[ 2a + 26d \bigg]$

$\displaystyle \: S_{28} = \frac{28}{2} \bigg[ 2a + (28 - 1)d \bigg]$

$\implies \: \displaystyle \: S_{28} = 14 \bigg[ 2a + 27d \bigg] = 28a + 378d$

Hence

$S_{28} - 2 \times S_{27} + S_{26}$

$\displaystyle \: = 28a + 378d - 27(2a + 26d) +( 26a + 325d)$

$= 54a + 703d - 54a - 702d$

$= d$

RESULT

$\sf \: {S_{28} - 2 \times S_{27} + S_{26} = common \: difference \: of \: the \:AP \: }$