Proof by induction: Base case: trynna see if the sequence is increasing. n=1 Xn € Xn+1 by definition X₁ = × 1+1 VEN x₁ = X2 √ ≤ 2 ✓ Induction Steps: Let n = k Э KEIN we know to show that and xx = Xxxl we are trying Xx+1 £ xk+2 Therefore, +2 (πx) = (xx+1)+3 √2+x=12+ X1 Xx+1 = Xk+2 Thus we can conclude that the sequence {X} Sequence. by the monotone definition is an increasing Now problem we know we need to show if it 15 bounded in the X ≤ 2 and given √2 = Proof by Induction: Base case: we know √2 ≤ X ≤ 2, UnEN √2 ± X₁ = 2 √2 ± √2 ± 2 ✓ Induction Steps: Let n = k B KEN ≤ and we want to we know √2 = Xx 2 Show that So. Xx+ 1 ≤ 2. +2 (√2)+2(x)(2) +2 √2+82 €12+ x =√4 2≤ Укн ≤2 Therefore sequence {Xx} In conclusion n In base on this we know that the is bounded. Show that the sequence {K^} is Therefore has the definition we d bounded In conclusion increasing and ' we show that the sequence 9x] 15 and bounded. Therefore by the definition of monotone convergence {v} is converges. thereom this sequence Now we need to find the limit: * Let X" be the limit lum (date) = √2+ km (x) 1im (xn+1) 7*2 +#2 - χ 2 2+ 2 + x* x-2=0 (x*+1) (x* -2) X=-1 = 0 x = 2 ** The limit is 2 2. Let the sequence (xn) be recursively defined by x₁ = √√2 and Xn+1 = √2+xn, n≥1. Show that (xn) converges and evaluate its limit.

Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a Tl-84, do the following. • In the home screen, press [2ND] LIST. • Scroll over to OPS and choose ‘seq(” from the dropdown list. Press [ENTER] • In the line headed “Expr:” type in the explicit formula, using the [X,T,,n] button for n• In the line headed ‘Variable” type In the variable used on the previous step. • In the line headed ‘start:” key in the value of n that begins the sequence. • In the line headed “end:’ key in the value of nthat ends the sequence. • Press [ENTER] 3 times to return to the home screen. You will see the sequence syntax on the screen. Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. Using a TI-83, do the following. • In the home screen, press [2ND] LIST. • Scroll over to OPS and choose “seq(“ from the dropdown list. Press [ENTER]. • Enter the items in the order “Expr’, Variable’, ‘start”. end separated by commas. See the instructions above for the description of each item. • Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary. 63. List the first six terms of the sequence. an=n33.5n2+4.1n1.52.4n