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The sum of the first 76 terms of an arithmetic progression (AP) is 21,850, and the sum of the first 40 terms is 7,900. Find the sum of the first 100 terms of this AP.

Answer :

Let's solve the problem using the information given about the arithmetic progression (AP).

We know:
- The sum of the first 76 terms is 21,850.
- The sum of the first 40 terms is 7,900.

We can use the formula for the sum of the first [tex]\( n \)[/tex] terms of an AP:

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

where [tex]\( S_n \)[/tex] is the sum of the first [tex]\( n \)[/tex] terms, [tex]\( a \)[/tex] is the first term, [tex]\( d \)[/tex] is the common difference, and [tex]\( n \)[/tex] is the number of terms.

### Step 1: Set up the equations.

Equation for 76 terms:

[tex]\[
S_{76} = \frac{76}{2} \times (2a + 75d) = 21,850
\][/tex]

Simplify it:

[tex]\[
38 \times (2a + 75d) = 21,850
\][/tex]

So:

[tex]\[
2a + 75d = 575
\][/tex]

Equation for 40 terms:

[tex]\[
S_{40} = \frac{40}{2} \times (2a + 39d) = 7,900
\][/tex]

Simplify it:

[tex]\[
20 \times (2a + 39d) = 7,900
\][/tex]

So:

[tex]\[
2a + 39d = 395
\][/tex]

### Step 2: Solve the system of equations.

Now, solve the two equations:

1. [tex]\( 2a + 75d = 575 \)[/tex]
2. [tex]\( 2a + 39d = 395 \)[/tex]

Subtract the second equation from the first to eliminate [tex]\( 2a \)[/tex]:

[tex]\[
(2a + 75d) - (2a + 39d) = 575 - 395
\][/tex]

This simplifies to:

[tex]\[
36d = 180
\][/tex]

Thus, the common difference [tex]\( d \)[/tex] is:

[tex]\[
d = \frac{180}{36} = 5
\][/tex]

### Step 3: Find the first term [tex]\( a \)[/tex].

Substitute [tex]\( d = 5 \)[/tex] back into one of the equations, for instance, the second one:

[tex]\[
2a + 39 \times 5 = 395
\][/tex]

[tex]\[
2a + 195 = 395
\][/tex]

[tex]\[
2a = 395 - 195
\][/tex]

[tex]\[
2a = 200
\][/tex]

So, the first term [tex]\( a \)[/tex] is:

[tex]\[
a = \frac{200}{2} = 100
\][/tex]

### Step 4: Find the sum of the first 100 terms.

Now, use the sum formula for 100 terms:

[tex]\[
S_{100} = \frac{100}{2} \times (2a + 99d)
\][/tex]

Substitute [tex]\( a = 100 \)[/tex] and [tex]\( d = 5 \)[/tex]:

[tex]\[
S_{100} = 50 \times (2 \times 100 + 99 \times 5)
\][/tex]

[tex]\[
S_{100} = 50 \times (200 + 495)
\][/tex]

[tex]\[
S_{100} = 50 \times 695
\][/tex]

[tex]\[
S_{100} = 34,750
\][/tex]

Therefore, the sum of the first 100 terms of the AP is 34,750.

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