Calculate the value of the following expression:

3∑i=1i∑j=1(2×i+3×j)\displaystyle \sum_{i=1}^3 \sum_{j=1}^i (2 \times i + 3 \times j)

Calculate the value of the following expression:

3∑i=13∑j=i(2×i+3×j)\displaystyle \sum_{i=1}^3 \sum_{j=i}^3 (2 \times i + 3 \times j)

Suppose a sequence is defined as:

a0a_0 = 3

ai=2×ai−1+3a_i = 2 \times a_{i-1} + 3 for all i≥1i \geq 1

Determine aia_i when ii is 3.

Suppose a sequence is defined as:

a0a_0 = 9

ai=2×ai−1+5a_i = 2 \times a_{i-1} + 5 for all i≥1i \geq 1

Determine aia_i when ii is 3.

Choose the main technique to prove the following statement:

3x+21=03x+21=0 has an integer solution.

Choose the main technique to prove the following statement:

x2+2x−8=0x^2 + 2x -8=0 has integer solutions.

Choose the main technique to prove the following statement.

√3\sqrt{3} is irrational.

Choose the main technique to prove the following statement.

If nn is even, then 5n2+6n+75n^2+6n+7 is odd.

Consider a function f:R→Rf:R→Rf: R \to R, f(x)=2x+8f(x)=2x+8f(x) = 2x+8.

Determine the correct sequence to prove the following statement. 

ff is surjective.

If a statement is not used in the proof, you have to choose "Not used".