The sum of an infinite geometric series is 12, and the square of the first term equals 4 times the product of the first term and the common ratio. What is the first term? - Treasure Valley Movers

The Sum of an Infinite Geometric Series Is 12—And the First Term’s Hidden Pattern Reveals Powerful Math

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Understanding the Problem: The Sum and the Pattern

We begin with the sum of an infinite geometric series:
[ S = \frac{a}{1 - r} = 12 ]
where ( a ) is the first term, ( r ) is the common ratio, and ( |r| < 1 ) ensures convergence.

From this, we isolate:
[ a = 12(1 - r) ]

The second condition states:
[ a^2 = 4ar ]

Dividing both sides by ( a ) (assuming ( a \ne 0 )) gives:
[ a = 4r ]

Now combine the two:
[ 4r = 12(1 - r) ]
Expand and solve:
[ 4r = 12 - 12r ]
[ 16r = 12 ]
[ r = \frac{3}{4} ]

Substitute back:
[ a = 4