Question

For a given arithmetic series the sum of the first 50 terms is 200. The sum of the next 50 terms is 2700. Find the first term of the series.

Answer 1

-20.5

Explanation:

The formula for the sum of the first n terms of an arithmetic series is:

`\boxed{S_n=(n)/(2)[2a+(n-1)d]}`

where:

• a is the first term.
• d is the common difference.
• n is the position of the term.

For a given arithmetic series, the sum of the first 50 terms is 200.

Therefore:

• n = 50
• Sₙ = 200

Substitute these values into the formula and rearrange to isolate d:

`\begin{aligned}(50)/(2)\left[2a+(50-1)d\right]&=200\\\\25\left[2a+49d\right]&=200\\\\2a+49d&=8\\\\49d&=8-2a\\\\d&=(8-2a)/(49)\end{aligned}`

The sum of the next 50 terms is 2700. Therefore, the sum of the first 100 terms is the sum of the first 50 terms and the next 50 terms.

Therefore:

• n = 50 + 50 = 100

• Sₙ = 200 + 2700 = 2900

Substitute these values into the formula and rearrange to isolate d:

`\begin{aligned}(100)/(2)\left[2a+(100-1)d\right]&=2900\\\\50\left[2a+99d\right]&=2900\\\\2a+99d&=58\\\\99d&=58-2a\\\\d&=(58-2a)/(99)\end{aligned}`

To find the first term of the series, substitute the second equation for d into the first equation for d and solve for a:

`\begin{aligned}(58-2a)/(99)&=(8-2a)/(49)\\\\49(58-2a)&=99(8-2a)\\\\2842-98a&=792-198a\\\\100a&=-2050\\\\a&=-20.5\end{aligned}`

Therefore, the first term of the series is a = -20.5.