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Sum of Infinite Geometric Sequence Equals 1/3 Look at this sequence: 1/4, 1/16, 1/64, 1/256, What do you think is the value when adding up the terms of this sequence? The surprising answer is 1/3.  If you find this answer surprising indeed, read on to figure out how this came to be. So our sum is  1/4 + 1/16 + 1/64 + 1/256 +  = 1/3 The sequence we deal with here is a  geometric  sequence. In a geometric  sequence  each term is found by  multiplying  the previous term by a  constant. In General  you could write a Geometric Sequence like this: {a,  ar,  ar2,  ar3, } Note, each term of our sequence can be computed with this rule:  xn  =  ar(n-1). Also note this sequence goes to infinity, . There exists a formula to compute the value of infinite sequences: Filling in the values, a=1 and r=1/4, the sum of our infinite sequence is = 1/(1-1/4) = 4/3 Because we have dropped the first value of this sequence (1/4)0 = 1 we have to subtract it from the result to reach the final answer 4/3 1 = 1/3. The picture visualizes our sequence (1/4, 1/16, ) as the green area in a square.  Do you also feel the sum of the green area makes about 1/3 of the squares entire area? For more help with sequences get a personal math tutor.