To solve the problem of finding the sum of a sequence where each term after the first increases by 1, we use the formula for the sum of an arithmetic sequence. Here's the step-by-step solution:
Key Observations
The sequence is an arithmetic sequence with a common difference of 1. For such sequences:
- Let the first term be (a_1), the last term be (a_n), and the number of terms be (n).
- The number of terms (n = a_n - a_1 + 1) (since each term increases by 1).
- The sum (S) of the sequence is given by:
[ S = \frac{n(a_1 + a_n)}{2} ]
Example Application
Suppose the sequence is (1,2,3,...,100):
- (a_1 = 1), (a_n = 100), (n = 100 -1 +1 =100).
- Sum: (S = \frac{100(1+100)}{2} = 5050).
General Answer
For any sequence starting at (a_1) and ending at (a_n) (each term +1), the sum is:
[
\boxed{\frac{(a_n - a_1 +1)(a_1 + a_n)}{2}}
]
If the sequence starts at 1 and has (k) terms (ends at (k)), the sum simplifies to:
[
\boxed{\frac{k(k+1)}{2}}
]
Note: If the original problem had specific values (e.g., first term=3, last term=7), substitute into the formula to get the exact sum (e.g., (\frac{(7-3+1)(3+7)}{2}=25)). For common cases like 1-100, the answer is 5050.
Assuming the most common scenario (sum from 1 to 100), the answer is:
(\boxed{5050})
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