Understanding the Arithmetic Progression Problem

The question involves an Arithmetic Progression (AP), which is a sequence of numbers such that the difference between consecutive terms is constant. This constant difference is called the common difference.

We are given a specific sum of eight terms in the AP:

We need to find the value of another expression involving some of these terms:

Method 1: Using the General Term Formula of AP

The general term of an AP is given by the formula:

where \(a\) is the first term (\(a_1\)) and \(d\) is the common difference.

Let's write out each term in the given sum using this formula:

• \(a_1 = a + (1-1)d = a\)
• \(a_5 = a + (5-1)d = a + 4d\)
• \(a_{10} = a + (10-1)d = a + 9d\)
• \(a_{15} = a + (15-1)d = a + 14d\)
• \(a_{20} = a + (20-1)d = a + 19d\)
• \(a_{25} = a + (25-1)d = a + 24d\)
• \(a_{30} = a + (30-1)d = a + 29d\)
• \(a_{34} = a + (34-1)d = a + 33d\)

Substitute these into the given sum equation:

Combine the terms:

Summing the coefficients of \(d\): \(4+9=13, 13+14=27, 27+19=46, 46+24=70, 70+29=99, 99+33=132\).

Dividing the entire equation by 4:

This equation relates the first term (\(a\)) and the common difference (\(d\)).

Now consider the expression we need to evaluate:

Substitute the general terms:

Expand and group terms with \(a\) and \(d\), paying close attention to the signs:

Collect terms with \(a\):

Collect terms with \(d\):

So the expression \(E\) simplifies to:

Method 2: Using a Property of Arithmetic Progressions

In an Arithmetic Progression, there is a useful property: if the sum of the indices of two terms is equal to the sum of the indices of two other terms, then the sum of those pairs of terms is also equal. That is, if \(i+j = k+l\), then \(a_i + a_j = a_k + a_l\).

Let's look at the indices of the terms in the given sum: 1, 5, 10, 15, 20, 25, 30, 34.

Consider pairs of indices:

• \(1 + 34 = 35\)
• \(5 + 30 = 35\)
• \(10 + 25 = 35\)
• \(15 + 20 = 35\)

Since the sum of indices for these pairs is the same (35), the sums of the corresponding terms are equal:

Let's denote this common sum by \(K\). So, \(K = a_1 + a_{34} = a_5 + a_{30} = a_{10} + a_{25} = a_{15} + a_{20}\).

The given sum is:

We can group the terms in pairs with index sums equal to 35:

Substitute \(K\) for each pair sum:

So, each pair sum is equal to 75. For example, \(a_1 + a_{34} = 75\), \(a_5 + a_{30} = 75\), etc.

Now consider the expression we need to evaluate:

Rearrange and group the terms carefully, keeping the signs correct:

Substitute the value \(K=75\) for each grouped pair sum:

Conclusion

Both methods show that the value of the expression \(a_1 + a_5 - a_{10} - a_{15} - a_{20} - a_{25} + a_{30} + a_{34}\) is 0.

Arithmetic Progression Problem Revision

Additional Information on Arithmetic Progressions (AP)

An Arithmetic Progression (AP) is a type of sequence where the difference between any two consecutive terms is constant. This constant difference is called the common difference, usually denoted by \(d\).

• Defining Property: \(a_{n+1} - a_n = d\) for all \(n \ge 1\).
• First Term (\(a_1\)): The initial term of the sequence. Sometimes denoted just by \(a\).
• Common Difference (\(d\)): The constant value added to get the next term.
• General Term (\(a_n\)): The formula to find the \(n\)-th term is \(a_n = a_1 + (n-1)d\).
• Sum of First \(n\) Terms (\(S_n\)): The sum of the first \(n\) terms can be found using \(S_n = \frac{n}{2}(a_1 + a_n)\) or \(S_n = \frac{n}{2}(2a_1 + (n-1)d)\).
• Symmetric Property: In a finite AP with \(N\) terms, the sum of terms equidistant from the beginning and end is constant: \(a_k + a_{N-k+1} = a_1 + a_N\).
• General Pair Property: As used in this problem, if \(i+j = k+l\), then \(a_i + a_j = a_k + a_l\). This holds true whether the terms are within a finite sequence or part of an infinite AP.

These properties are helpful in solving problems involving sums or relationships between non-consecutive terms in an Arithmetic Progression.