Geometric seies

The Proof

• Sn = a(1-r^n)/1-r
• Sn = a + ar +ar^2 + ar^3 + . . . + ar^(n-2) + ar^(n-1) (1)
• rSn = ar + ar^2 + ar^3 + ar^4 + . . . + ar^(n-1) + ar^n (2) (Multiply everything by r)
• Sn - rSn = a - ar^2 ((1) - (2))
• (1-r)Sn = a(1-r^n) (Factorise)
• Sn = a(1-r^n)/1-r

Properties

• Progression found by multiplying previous term by a non-zero number (common ratio)
• a - first term
• n - number of terms
• r - common ratio
• Divergent
• If the common ratio |r|≥ 1
• Sequence increases to ∞
• Convergent
• Tends towards a small value
• |r|< 1
• Finding a value > Un = arⁿ⁻1