(6) (bonus) here is a proof that for n > 0, 12 22 +..+2" = 2n+1. proof. suppose 1 + 2 + 2 +..+ 2" = 2n+1 for some n > 0. then 2n+12n+1 .. +2" +2"n+1. 1222 using the inductive hypothesis 2(2"+1) 2"+2 2(n+1)+1 as we needed to show. now, obviously there is something wrong with this proof by induction since, for example, 1+222 7, but 22+1 = 23 = 8. what specifically is wrong with the proof?

Intersecting secant-tangent theorem states that if a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.so to break it down: 1. pq acts as the tangent and ps as the secant with an intersection at p.sr = 21rp = 3x + 3pq = 4x + 4following to equation: tangent squared = the two segments multiplied by each otherso (4x + 4) ^ 2 = (3x + 3) • (21) simplifies to 16x^2 - 31x - 47 = 0 or (16x - 47)(x + 1) x = -1 or x = 47/162. pq acts as the tangent and ps as the secant with an intersection at p (again! )sr = 16rp = xpq = 15(15) ^ 2 = 16 • x225 = 16xx = 225/163. the interesting chords are proportional to each other so a ratio is possible to set up: ap dp 10 3 + x = = —— = pc pb 8 xcross multiple to get 10x = (3 + x)(8)x = 12

ℓσℓ ιт ∂σит тяσυвℓє мє αт αℓℓ ιм gℓα∂ тσ нєℓρ ρρℓ συт

ωє ιт ¢αи вє ωяιттєи αѕ

тнє єχα¢т fσям: 89/9тнє ∂є¢ιмαℓ fσям: 9.ιт яєαρєαтѕ)αи∂ тнє мιχє∂ иυмвєя fσям 9 8/9

step-by-step explanation:

нσρє ι ¢συℓ∂ нєℓρ уσυ συт