Suppose a sequence an, is defined as follows: a1 = 9/10, a2 = 10/11, an+2 = an+1an.Show that 0 < an < 1 for all (positive integers) n.

Suppose a sequence an, is defined as follows:a1 = 9/10, a2 = 10/11, an+2 = an+1an

Let (an) be a sequence defined bya1 = 12, an+1 = 2 − 2an + 1 for n ≥ 1.(a) Show that an ∈ [0, 1] for all n. [4 marks](b) Show that (an) is increasing. [5 marks](c) Show that (an) is convergent, and calculate its limit.

. If the nth term of the sequence < an > is given by"211 322 433· ·n + 1nn# 1nFind the limit of sequence

You are given a sequence [a1,…,an][𝑎1,…,𝑎𝑛] where each element ai𝑎𝑖 is either 00 or 11. You can apply several (possibly zero) operations to the sequence. In each operation, you select two integers 1≤l≤r≤|a|1≤𝑙≤𝑟≤|𝑎| (where |a||𝑎| is the current length of a𝑎) and replace [al,…,ar][𝑎𝑙,…,𝑎𝑟] with a single element x𝑥, where x𝑥 is the majority of [al,…,ar][𝑎𝑙,…,𝑎𝑟].Here, the majority of a sequence consisting of 00 and 11 is defined as follows: suppose there are c0𝑐0 zeros and c1𝑐1 ones in the sequence, respectively.If c0≥c1𝑐0≥𝑐1, the majority is 00.If c0<c1𝑐0<𝑐1, the majority is 11.For example, suppose a=[1,0,0,0,1,1]𝑎=[1,0,0,0,1,1]. If we select l=1,r=2𝑙=1,𝑟=2, the resulting sequence will be [0,0,0,1,1][0,0,0,1,1]. If we select l=4,r=6𝑙=4,𝑟=6, the resulting sequence will be [1,0,0,1][1,0,0,1].Determine if you can make a=[1]𝑎=[1] with a finite number of operations.InputEach test contains multiple test cases. The first line contains the number of test cases t𝑡 (1≤t≤4⋅1041≤𝑡≤4⋅104). Description of the test cases follows.The first line of each testcase contains one integer n𝑛 (1≤n≤2⋅1051≤𝑛≤2⋅105).The second line of each testcase contains a string consisting of 00 and 11, describing the sequence a𝑎.It's guaranteed that the sum of n𝑛 over all testcases does not exceed 2⋅1052⋅105.OutputFor each testcase, if it's possible to make a=[1]𝑎=[1], print YES. Otherwise, print NO. You can output the answer in any case (upper or lower). For example, the strings yEs, yes, Yes, and YES will be recognized as positive responses.ExampleinputCopy5101120191000000019000011000outputCopyNoYesNoYesNoNoteIn the fourth testcase of the example, initially a=[1,0,0,0,0,0,0,0,1]𝑎=[1,0,0,0,0,0,0,0,1]. A valid sequence of operations is:Select l=2,r=8𝑙=2,𝑟=8 and apply the operation. a𝑎 becomes [1,0,1][1,0,1].Select l=1,r=3𝑙=1,𝑟=3 and apply the operation. a𝑎 becomes [1][1].

A sequence is given by the formula of its nth term : an =10−3n. prove that an is an arithmetic progression.