Find the sums given below :
`7+10 1/2 +14 +…..+84`

Answer

To solve the problem of finding the sum of the series \(7 + 10 \frac{1}{2} + 14 + \ldots + 84\), we will follow these steps:

### Step 1: Identify the first term and common difference
The first term \(a\) of the series is \(7\). The second term is \(10 \frac{1}{2}\), which can be converted to an improper fraction:
\[ 10 \frac{1}{2} = \frac{21}{2} \]
Now, we can find the common difference \(d\) by subtracting the first term from the second term:
\[ d = \frac{21}{2} - 7 = \frac{21}{2} - \frac{14}{2} = \frac{7}{2} \]

### Step 2: Identify the last term
The last term of the series is given as \(84\).

### Step 3: Use the formula for the nth term of an arithmetic progression
The formula for the nth term of an arithmetic progression is given by:
\[ a_n = a + (n-1)d \]
Setting \(a_n = 84\), we can solve for \(n\):
\[ 84 = 7 + (n-1) \cdot \frac{7}{2} \]
Subtract \(7\) from both sides:
\[ 77 = (n-1) \cdot \frac{7}{2} \]
Multiply both sides by \(2\) to eliminate the fraction:
\[ 154 = (n-1) \cdot 7 \]
Now, divide by \(7\):
\[ n-1 = 22 \]
Thus, adding \(1\) to both sides gives:
\[ n = 23 \]

### Step 4: Calculate the sum of the series
The sum \(S_n\) of the first \(n\) terms of an arithmetic progression can be calculated using the formula:
\[ S_n = \frac{n}{2} \cdot (a + l) \]
where \(l\) is the last term. Substituting the values we have:
\[ S_{23} = \frac{23}{2} \cdot (7 + 84) = \frac{23}{2} \cdot 91 \]
Calculating this gives:
\[ S_{23} = \frac{23 \cdot 91}{2} = \frac{2093}{2} = 1046.5 \]

### Final Answer
The sum of the series \(7 + 10 \frac{1}{2} + 14 + \ldots + 84\) is \(1046.5\).

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