<b>Setup:</b><br>Charges of magnitude q are placed at:<br>x = 1, 2, 4, 8, 16, ... (powers of 2)<br><br>We need to find E at x = 0.<br><br><b>Step 1: Field due to each charge</b><br><br>E₁ (at x=1, distance=1) = kq/(1)² = kq<br>E₂ (at x=2, distance=2) = kq/(2)² = kq/4<br>E₃ (at x=4, distance=4) = kq/(4)² = kq/16<br>E₄ (at x=8, distance=8) = kq/(8)² = kq/64<br>... and so on.<br><br><b>Step 2: All fields point in same direction</b><br>All charges are on positive x-axis, so all fields at origin point along negative x-direction (if q is positive).<br><br><b>Step 3: Sum up using geometric series</b><br>E_total = kq[1 + 1/4 + 1/16 + 1/64 + ...]<br><br>This is a <b>geometric progression</b> with:<br>• First term a = 1<br>• Common ratio r = 1/4<br><br><b>Step 4: Apply GP sum formula</b><br>Sum = a/(1 - r) = 1/(1 - 1/4) = 1/(3/4) = <b>4/3</b><br><br><b>Final Answer:</b><br><b>E = (4/3) × kq = 4kq/3</b><br><br>Or equivalently: E = q/(3πε₀)