25 + 26 + .... + 75 = ?
The problem asks for the sum of the series 25 + 26 + ... + 75. This is an arithmetic progression.
Identifying Series Parameters
- First term, $a = 25$.
- Last term, $l = 75$.
- Common difference, $d = 1$ (since each term increases by 1).
Calculating the Number of Terms (n)
Use the formula for the $n$-th term of an arithmetic progression: $l = a + (n-1)d$.
Substitute the known values:
$75 = 25 + (n-1) \times 1$
Solve for $n$:
$75 - 25 = n - 1$
$50 = n - 1$
$n = 50 + 1 = 51$
There are 51 terms in the series.
Calculating the Sum of the Series (S)
Use the formula for the sum of an arithmetic progression: $S_n = \frac{n}{2}(a + l)$.
Substitute the values of $n$, $a$, and $l$:
$S_{51} = \frac{51}{2}(25 + 75)$
$S_{51} = \frac{51}{2}(100)$
$S_{51} = 51 \times 50$
$S_{51} = 2550$
The sum of the series is 2550.
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Important Questions from Arithmetic Progression
If the nth term of a sequence is \(\frac{2 n+5}{7}\), then what is the sum of its first 140 terms?
What is the arithmetic mean of first 8 multiples of 13?
The average of five consecutive odd natural numbers is 27. The product of the first and fifth number is:
Find the sum of all the numbers between 100 to 200 which are divisible by 12.
In a garden, there are 6 daisy plants the first year. Each year, a gardener adds 3 new daisy plants the first year and loses 2 each year. He has 26 jasmine plants the first year and loses 2 each year. When will the number of daisy plants equal the number of jasmine plants after the first year?
