For an AP(arithmetic progression) if S49=100 and T50=25 then S50=____

The given $T 50 = 20$

and $S 49 = 100$

We know that the formula  $Tn = a +n(-1)d \\$ and $Sn = \frac{n}{2} [2a + (n-1)d] \\$

then

$T50 = 25\\ a + 49 d = 25\\ a = 25 -49d ......(1)$

and

$Sn = 100 \\ \frac{49}{2}[2(25 - 49d) + 48d] = 100 \\ 49[50 -98d + 48d] = 200\\ 2450 - 2450 d =200\\ d = \frac{45} {49}$

Now putting this value in Equation (1)

$a = 25 - \frac{45}{49} \times 49 \\ a = 25 -45 \\ a = -20$

Now                 $S50 = \frac{50}{2} [-40 + 45]\\ S50 = 125$

Hence, $S50 = 125 \\$

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