So i have the following sequence: 1,1,2,2,2,3,4,5,5,4,4,4,4,4,5,… I would like to know the n-th term of this sequence.

Question

So i have the following sequence:  1,1,2,2,2,3,4,5,5,4,4,4,4,4,5,…  I would like to know the n-th term of this sequence.

coolbananas03ok · Accepted Answer

If tn denotes the value of the n-th term then:  2+3+…+tn&amp;lt;n≤2+3+…+tn+(tn+1)  =12tn(tn+3)  so that tn is the smallest integer that satisfies:  2n≤tn(tn+3)  leading to tn=[129+8n−32]

goleuedigdp · Accepted Answer

n∈N appears n+1 times starting from the position  1+∑i=1n−1(i+1)=n+n(n+1)2=12n(n+3)  Therefore,  an=k⇔n∈[12k(k+3),k+12k(k+3)]  so k is the smallest integer with n≥12k(k+3)  Solving this for k, we obtain  an=[−3+9+8n2], n∈N

RizerMix · Accepted Answer

Given 1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,5,..., first note that: αT(k)=k where T(k) is a triangular number. Indeed: aT(1)=a1=1; aT(2)=a3=2; aT(3)=a6=3; aT(4)=a10=4; aT(5)=a15=5... Hence: T(k)≤n≤T(k+1)−1, an=k⇔k(k+1)2≤n≤(k+1)(k+2)2−1 ⇔k2+k−2n≤0≤k2+3k−2n ⇔[8n+9−32]≤k≤[8n+1−12] Hence: an=8n+9−32 or 8n+1−12

So i have the following sequence: 1,1,2,2,2,3,4,5,5,4,4,4,4,4,5,... I would like to know

Answered question

Answer & Explanation

New Questions in Calculus 2