The second term of a geometric series is -2/3 and its sum to infinity is 3/2. Find its common ratio.

Step 1: Write formulas for G.P.
Second term: ar = -2/3
Sum to infinity: a/(1 – r) = 3/2
Where a = first term, r = common ratio

Step 2: Express a from sum formula
a/(1 – r) = 3/2
a = (3/2)(1 – r)

Step 3: Substitute into second term equation
ar = -2/3
[(3/2)(1 – r)] × r = -2/3
(3/2)r – (3/2)r² = -2/3

Step 4: Multiply through by 2
3r – 3r² = -4/3
Multiply by 3: 9r – 9r² = -4
9r² – 9r – 4 = 0

Step 5: Solve quadratic using formula
r = [9 ± √(81 + 144)]/18
r = [9 ± √225]/18
r = [9 ± 15]/18
r = 24/18 = 4/3 or r = -6/18 = -1/3

Step 6: Check which r gives convergent series
For convergence: |r| < 1
|4/3| = 4/3 > 1 (diverges)
|-1/3| = 1/3 < 1 (converges ✓)
Therefore r = -1/3