Write \(\sum_{i=1}^{4} i\) without sigma notation.

In \(\sum_{k=3}^{8} a_k\), identify the index, lower bound, upper bound, and term.

Evaluate \(\sum_{i=1}^{3} 5\).

Expand \(\sum_{j=0}^{3} 2^j\).

Evaluate \(\sum_{i=1}^{5} i\) using the formula for the first \(n\) integers.

Evaluate \(\sum_{i=1}^{4} i^2\).

Evaluate \(\sum_{i=1}^{4}(2i+1)\).

Rewrite \(4+7+10+13\) using sigma notation with index \(i\) starting at \(1\).

Use summation rules to evaluate \(\sum_{i=1}^{6}(3i-2)\).

Evaluate the geometric sum \(\sum_{i=0}^{4} 3\cdot2^i\).

Explain why \(\sum_{i=1}^{n} c=nc\) for a constant \(c\).

A physics model samples positions \(x_i\) at four equal times. Write the total sampled displacement contribution \(2x_1+2x_2+2x_3+2x_4\) in sigma notation and factor the constant.

Evaluate \(\sum_{i=1}^{5}(i^2+i)\).

Evaluate \(\sum_{i=2}^{5} i^2\) without changing the index bounds incorrectly.

Find \(n\) if \(\sum_{i=1}^{n} i=45\).

Evaluate \(\sum_{i=1}^{n}(2i-1)\) in closed form.

Evaluate \(\sum_{i=1}^{3}\sum_{j=1}^{2}(i+j)\).

Diagnose the error: \(\sum_{i=1}^{4} i^2=(\sum_{i=1}^{4}i)^2\). Give the correct values.

Show that changing the dummy index name does not change the value of \(\sum_{i=1}^{4} i^2\).

For \(|r|<1\), evaluate \(\sum_{i=0}^{\infty} 6\left(\frac13\right)^i\) and explain why the finite formula would not be enough by itself.