The sum of the 4th and the 14th term of an arithmetic sequence equals 22. The 9th term is 4 larger than the 4th term. Calculate the first term of the sequence.
I've been trying to solve this for like an hour but I just can't figure out how to do it! I'd really appreciate your help! Thanks!!
"The sum of the 4th and the 14th term of an arithmetic sequence equals 22" ----> a+3d + a+13d = 22
2a + 16d = 22
a + 8d = 11 (#1)
"The 9th term is 4 larger than the 4th term" ---> a+8d - (a+3d) = 4
5d = 4
d = 4/5
a = 11-8d
a = 11 - 8(4/5) = 23/5
check:
t4 = 23/5 + 3(4/5) = 35/5 = 7
t9 = 23/5 + 8(4/5) = 55/5 = 11
t14 = 23/5 + 13(4/5) = 75/5 = 15
11+15 = 22
11-7=4
all is good
The third term of an arithmetic sequence is 15, and the fifth term is 37. What is the first term?
To solve this problem, we can use the properties of arithmetic sequences. Let's break it down step by step:
Step 1: Define the terms of the arithmetic sequence.
Let's assume the first term of the sequence is denoted as "a" and the common difference between the terms is denoted as "d."
Step 2: Determine the formulas for the terms of the sequence.
The formula for the nth term of an arithmetic sequence is given by: Tn = a + (n - 1)d
Step 3: Set up equations using the given information.
We're given two pieces of information:
- The sum of the 4th and 14th term equals 22: T4 + T14 = 22
- The 9th term is 4 larger than the 4th term: T9 = T4 + 4
Step 4: Substitute the formulas into the equations.
Using the formula for the nth term from step 2, we can rewrite the equations as:
- a + 3d + a + 13d = 22
- a + 8d = a + 3d + 4
Step 5: Simplify and solve the equations.
Simplifying the equations further, we get:
- 2a + 16d = 22
- 5d = 4
Step 6: Solve for the variables.
Solving the second equation, we find that d = 4/5 = 0.8.
Substituting this value of d into the first equation, we can solve for a:
2a + 16(0.8) = 22
2a + 12.8 = 22
2a = 9.2
a = 4.6
Therefore, the first term of the sequence is 4.6.