Let a_1, a_2, . . . , a_10 be an arithmetic sequence. If a_1 + a_3 + a_5 + a_7 + a_9 = 17 and a_2 + a_4 + a_6 + a_8 + a_{10} = 15, then find a_1.
Stop Cheating!
@AoPS Ur a cheater pretending to be an AoPS staff member
To find the value of a₁, we can use the given information to set up a system of equations.
In an arithmetic sequence, the common difference between any two consecutive terms is constant. Let's denote this common difference as "d".
We know that a₁ + a₃ + a₅ + a₇ + a₉ = 17. Let's express these terms in terms of a₁ and d:
a₃ = a₁ + 2d
a₅ = a₁ + 4d
a₇ = a₁ + 6d
a₉ = a₁ + 8d
Substituting these values back into the equation, we have:
a₁ + (a₁ + 2d) + (a₁ + 4d) + (a₁ + 6d) + (a₁ + 8d) = 17
Simplifying this equation, we get:
5a₁ + 20d = 17 --(Equation 1)
Similarly, we know that a₂ + a₄ + a₆ + a₈ + a₁₀ = 15.
a₂ = a₁ + d
a₄ = a₁ + 3d
a₆ = a₁ + 5d
a₈ = a₁ + 7d
a₁₀ = a₁ + 9d
Substituting these values back into the equation, we have:
(a₁ + d) + (a₁ + 3d) + (a₁ + 5d) + (a₁ + 7d) + (a₁ + 9d) = 15
Simplifying this equation, we get:
5a₁ + 25d = 15 --(Equation 2)
Now, we have a system of two equations (Equation 1 and Equation 2) with two unknowns (a₁ and d). We can solve this system to find the value of a₁.
Let's solve these equations:
Multiply Equation 1 by 5, and Equation 2 by -4 to eliminate the variable 'd':
25(5a₁ + 20d) - 4(5a₁ + 25d) = 17(5) - 15(4)
125a₁ + 500d - 20a₁ - 100d = 85 - 60
105a₁ + 400d - 100d = 25
105a₁ + 300d = 25 --(Equation 3)
Now, let's solve Equation 3 for a₁:
105a₁ + 300d = 25
105a₁ = 25 - 300d
a₁ = (25 - 300d)/105
Therefore, the value of a₁ is given by a₁ = (25 - 300d)/105.
To find the specific value of a₁, we need to know the value of 'd' (the common difference) or have more information.
Please provide the value of 'd' or any additional information to find the specific value of a₁.
5/2 (a + a+4*2d) = 17
5/2 (a+d + a+d+4*2d) = 15
a=5, d = -2/5
Check: The odd terms are
5, 21/5, 17/5, 13/5, 9/5. Sum=17
and I bet the evens add up as well.