Find the sum of the first 777 terms of the arithmetic sequen

Find the sum of the first 777 terms of the arithmetic sequence: 3, 9, 15, 21,... Find the exact sum of the infinite geometric sequence. 1/2, -1/4, 1/8, hellip 32, -16, 8, -4, hellip 3, 2, 4/3, 8/9, hellip -54, -18, -6, -2, hellip

Solution

The formula for sum of n terms in Arithmetic Sequenceis

(n/2)(2u₁ + (n-1)d)

Here n = 777

u₁ = 3

d = 6

Sum = (777/2)(2*3 + (777-1)*6) = 1,811,187

The formula for sum of infinite Geometric Sequence is

S =u₁ /(1-r)

a) u₁ = 1/2

r = -1/2

S = (1/2)/(1- (-1/2))

S = (1/2)/(1+1/2)

S = (1/2)/(3/2)

S = (1/2)*(2/3) = 1/3

b) u₁ = 32

r = -1/2

S = 32/(1-(-1/2))

S = 32/(1+1/2)

S = 32/(3/2)

S = 32*(2/3) = 64/3

C) u₁ = 3

r = 2/3

S = 3/(1-2/3)

S = 3/(1/3)

S = 3*(3/1) = 9

d) u₁ = -54

r = 1/3

S = -54/(1-1/3)

S = -54/(2/3)

S = -54(3/2) = -81