Question

7. For each of the following, prove that the sequence {a,) converges and find the limit. 2. Qn+1 = (2a, + 5). a, = 2 b. a -1 = V2an, a, = 3 *c. Qn+1 = V2a, + 3. a, - 1 d. 2n+1 = V2a, + 3, a, = 4

Answer 1

please thumb up. Thanks

Thus the sequence is decreasing and bounded above. let to we get 100 100 - 2 Ans: o anti = Ź (29, +5] 9, = 2 ») 9 = ☆ (20,+5) - Ś (4+5) = 3 - 2 9,39,3979-.. go Ś (294+5) = Ć (x3 +5) = 24 od=(29,+5) = + (9 ) = 23 also the sequence is non-negatire. L be Thus connesgent. the limit or seg. then apply limit Jim (9+) = lim (( 29+s-) ). a = [2 (lima +57 - - [22+] = L. & 6 L=2Lts - 42=5 1 2 = 15 16 1 2) La 5 is the limit at given sequence, %*, Jean 9,= 3 9 - Jaan - Sans - Jo 90+ V2.0, sex to 94 120 B ravo This is a decreasing nuninegative sequence. let L = limit o a sequence. apply the limit ter ② 7 lim (anti) = lim (1290) = 2. lim an L-12L saugoing on both sides we get L=0 or L=2 ² = 22 - = L(L-2) = 0 22-2L = 0 [2=2 since 22 &n. © anti = √29, +3 ,91 =1. 9 = $2x 143 = 55, 9 = $ers to - 94 = t3 & 9, <a Laz.... thus given sequenea is increasing & bounded below. (non-negative) ler = limit of a sequence. apply limit to ③ we get a 2. liman+3 = lim (Anty) = lim Sants 2.2+3 - L - g ² = 2L+3 = 2²-2 L-350 L=34 L = -1 9 limit is =3 since sequence is non-negative nta n-