The sum of n terms of the series $1 + 1.5 + 2 + 2.5 + 3 + ...$ is?

The first few terms of the series are:

• 1st term = 1
• 2nd term = 1.5
• 3rd term = 2
• 4th term = 2.5
• 5th term = 3

The difference between consecutive terms:

• 1.5 - 1 = 0.5
• 2 - 1.5 = 0.5
• 2.5 - 2 = 0.5

Since the difference is constant (always 0.5), this is an Arithmetic Progression (AP).

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For this AP:

First term (a) = 1

Common difference (d) = 0.5

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The formula for sum of n terms of an AP is:

$S_n = \frac{n}{2}[2a + (n-1)d]$

Substituting the values:

$S_n = \frac{n}{2}[2(1) + (n-1)(0.5)]$

$S_n = \frac{n}{2}[2 + 0.5n - 0.5]$

$S_n = \frac{n}{2}[1.5 + 0.5n]$

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Taking out 0.5 as common from the bracket:

$S_n = \frac{n}{2} \times 0.5[3 + n]$

$S_n = \frac{n \times 0.5}{2}[3 + n]$

$S_n = \frac{n}{4}[n + 3]$

$S_n = \frac{n(n+3)}{4}$

Therefore, the sum of n terms of the series is $\frac{n(n+3)}{4}$.