To determine which sum is greater between [tex]\( O \)[/tex] and [tex]\( E \)[/tex], and by how much, we first need to remember that [tex]\( O \)[/tex] and [tex]\( E \)[/tex] are arithmetic sequences.
Step 1: Define the sequences
- The sequence [tex]\( O \)[/tex] has its first term [tex]\( a_O = 5 \)[/tex] and its common difference [tex]\( d_O = 2 \)[/tex]. The last term of [tex]\( O \)[/tex] is 107.
- The sequence [tex]\( E \)[/tex] has its first term [tex]\( a_E = 4 \)[/tex] and its common difference [tex]\( d_E = 2 \)[/tex]. The last term of [tex]\( E \)[/tex] is 106.
Step 2: Calculate the number of terms in each sequence
The number of terms [tex]\( n \)[/tex] in an arithmetic sequence is given by:
[tex]\[ n = \frac{\text{last term} - \text{first term}}{\text{common difference}} + 1 \][/tex]
For the sequence [tex]\( O \)[/tex]:
[tex]\[ n_O = \frac{107 - 5}{2} + 1 \][/tex]
[tex]\[ n_O = \frac{102}{2} + 1 \][/tex]
[tex]\[ n_O = 51 + 1 \][/tex]
[tex]\[ n_O = 52 \][/tex]
For the sequence [tex]\( E \)[/tex]:
[tex]\[ n_E = \frac{106 - 4}{2} + 1 \][/tex]
[tex]\[ n_E = \frac{102}{2} + 1 \][/tex]
[tex]\[ n_E = 51 + 1 \][/tex]
[tex]\[ n_E = 52 \][/tex]
Step 3: Calculate the sum of each sequence
The sum of an arithmetic series is given by:
[tex]\[ S = n \left( \frac{a + l}{2} \right) \][/tex]
where [tex]\( n \)[/tex] is the number of terms, [tex]\( a \)[/tex] is the first term, and [tex]\( l \)[/tex] is the last term.
For the sequence [tex]\( O \)[/tex]:
[tex]\[ S_O = 52 \left( \frac{5 + 107}{2} \right) \][/tex]
[tex]\[ S_O = 52 \left( \frac{112}{2} \right) \][/tex]
[tex]\[ S_O = 52 \times 56 \][/tex]
[tex]\[ S_O = 2912 \][/tex]
For the sequence [tex]\( E \)[/tex]:
[tex]\[ S_E = 52 \left( \frac{4 + 106}{2} \right) \][/tex]
[tex]\[ S_E = 52 \left( \frac{110}{2} \right) \][/tex]
[tex]\[ S_E = 52 \times 55 \][/tex]
[tex]\[ S_E = 2860 \][/tex]
Step 4: Compare the sums and find the difference
We can see that:
[tex]\[ S_O = 2912 \][/tex]
[tex]\[ S_E = 2860 \][/tex]
The sum of [tex]\( O \)[/tex] is greater than the sum of [tex]\( E \)[/tex].
To find the difference:
[tex]\[ \text{Difference} = S_O - S_E \][/tex]
[tex]\[ \text{Difference} = 2912 - 2860 \][/tex]
[tex]\[ \text{Difference} = 52 \][/tex]
Conclusion:
The sum of the sequence [tex]\( O \)[/tex] is greater than the sum of the sequence [tex]\( E \)[/tex] by 52.
