zi2lalZ

2021-08-14

Discrete Math
Solve the following.
a) List the first four terms of the recursive sequence defined by ${s}_{1}=1$ and $\mathrm{\forall }n\ge 2,{s}_{n}={\left({s}_{n-1}+1\right)}^{2}$
b) Given that $\sum _{n=1}^{n}i=\frac{n\left(n+1\right)}{2}$, find the sum $2+4+6+\dots +200$.

cyhuddwyr9

Step 1
a) It is given that ${s}_{1}=1$ and ${s}_{n}={\left({s}_{n-1}+1\right)}^{2}$ for all $n\ge 2$.
Evaluate the second term of the sequence as follows.
${s}_{2}={\left({s}_{2-1}+1\right)}^{2}$
$={\left({s}_{1}+1\right)}^{2}$
$={\left(1+1\right)}^{2}$
$=4$ Thus, the second term is 4.
Evaluate the third term of the sequence as follows.
${s}_{3}={\left({s}_{3-1}+1\right)}^{2}$
$={\left({s}_{2}+1\right)}^{2}$
$={\left(4+1\right)}^{2}$
$=25$
Thus, the third term is 25.
Evaluate the fourth term of the sequence as follows.
${s}_{4}={\left({s}_{4-1}+1\right)}^{2}$
$={\left({s}_{3}+1\right)}^{2}$
$={\left(25+1\right)}^{2}$
$=676$
Thus, the fourth term is 676.
Therefore, the required first four terms are 1, 4, 25 and 676.
Step 2
b) It is given that $\sum _{i=1}^{n}i=\frac{n\left(n+1\right)}{2}$
Evaluate the given sum as follows.
$2+4+6+\dots +200=2\left[1+2+3+\dots +100\right]$
$=2\left[\frac{100\left(100+1\right)}{2}\right]$ $\left(\sum _{i=1}^{n}i=\frac{n\left(n+1\right)}{2}\right)$
$=100×101$
$=10100$
Therefore, the required sum is 10,100.

Do you have a similar question?