Question

find the sums given below:
(1) 34+32+30+..........+10
(2) -5+(-8)+(-11)+..........+(-230)

Answer 1

$S_n = \frac{n}{2} [T_1 + T_n] \\ \\ n = \frac{T_n - T_1}{d} + 1$

n = no. of terms

d = common difference

1.

$d = 32 - 34 = - 2$

$T_1 = 34 \\ T_n = 10$

$n = \frac{10 - 34}{- 2} + 1 \\ \\ = \frac{- 24}{- 2} + 1 \\ \\ = 12 + 1 = 13$

$S_n = \frac{13}{2} [34 + 10] \\ \\ = \frac{13}{2} \times 44 = 13 \times 22 = 286$

2.

$d = (- 8) - (- 5) = - 8 + 5 = - 3$

$T_1 = - 5 \\ T_n = - 230$

$n = \frac{(- 230) - (- 5)}{- 3} + 1 \\ \\ = \frac{- 230 + 5}{- 3} + 1 \\ \\ = \frac{- 225}{- 3} + 1 \\ \\ = 75 + 1 = 76$

$S_n = \frac{76}{2} [(- 5) + (- 230)] \\ \\ = \frac{76}{2} [- 5 - 230] \\ \\ = \frac{76}{2} \times (- 235) = 38 \times (- 235) = - 8930$

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