Find the sum of 100 terms of the geometric sequence. Find the sum of 100 terms of the geometric sequence.  1, -1, 1, -1, 1, -1...

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jeew-m | College Teacher | (Level 1) Educator Emeritus

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First term(a or `T_1) = 1`

Second term`(T_2) = -1`

 

The common ratio 'r' of a geometric series is given by;

`r = T_(n+1)/T_n`

`r = T_2/T_1`

`r = -1/1`

`r = -1`

 

The sum of a geometric series is given by;

`S_n = (a(1-r^n))/(1-r)`

 

So the sum of 100 terms is given by;

`S_100 = (1(1-(-1)^100))/(1-(-1))`

`S_100 = (1(1-1))/(1+1)`

`S_100 = (1xx0)/2`

`S_100 = 0`

 

So the sum of 100 terms is 0.

 

Easy method

If you have one term sum is 1

If you have two terms sum is 1+(-1) = 0

If you have three terms sum is 1+(-1)+1 = 1

If you have four terms sum is 1+(-1)+1+(-1) = 0

 

So;

If the number of terms is odd the sum is 1.

If the number of terms is even sum is 0.

Since 100 terms is a even number sum is 0.

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