33. Find the first term and the common difference of the arithmetic sequence whose 7 th term is 11 and 17 th term is 26 .

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To find the first term and the common difference of an arithmetic sequence, we can use the formula for the nth term of an arithmetic sequence:

$$ a_n = a_1 + (n-1)d $$

where:
- $$ a_n $$ is the nth term of the sequence,
- $$ a_1 $$ is the first term of the sequence,
- $$ n $$ is the term number,
- $$ d $$ is the common difference.

Given that the 7th term is 11 and the 17th term is 26, we can set up two equations using the formula:

For the 7th term:
$$ 11 = a_1 + (7-1)d $$
$$ 11 = a_1 + 6d $$

For the 17th term:
$$ 26 = a_1 + (17-1)d $$
$$ 26 = a_1 + 16d $$

Now, we can solve these two equations simultaneously to find the values of $$ a_1 $$ and $$ d $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}11=a_{1}+6d\26=a_{1}+16d\end{array}\right.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}a_{1}=11-6d\26=a_{1}+16d\end{array}\right.$$
- step2: Substitute the value of $$a_{1}:$$
  $$26=11-6d+16d$$
- step3: Add the terms:
  $$26=11+10d$$
- step4: Swap the sides:
  $$11+10d=26$$
- step5: Move the constant to the right side:
  $$10d=26-11$$
- step6: Subtract the numbers:
  $$10d=15$$
- step7: Divide both sides:
  $$\frac{10d}{10}=\frac{15}{10}$$
- step8: Divide the numbers:
  $$d=\frac{3}{2}$$
- step9: Substitute the value of $$d:$$
  $$a_{1}=11-6\times \frac{3}{2}$$
- step10: Simplify:
  $$a_{1}=2$$
- step11: Calculate:
  $$\left\{ \begin{array}{l}a_{1}=2\d=\frac{3}{2}\end{array}\right.$$
- step12: Check the solution:
  $$\left\{ \begin{array}{l}a_{1}=2\d=\frac{3}{2}\end{array}\right.$$
- step13: Rewrite:
  $$\left(a_{1},d\right) = \left(2,\frac{3}{2}\right)$$
The solution to the system of equations is:
$$ a_1 = 2 $$
$$ d = \frac{3}{2} $$

Therefore, the first term of the arithmetic sequence is 2, and the common difference is $$ \frac{3}{2} $$.
The first term is 2 and the common difference is $$ \frac{3}{2} $$.

33. Find the first term and the common difference of the arithmetic sequence whose 7 th term is 11 and 17 th term is 26 .

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