the 10th term of an arithmetic sequence is equal to the sum of 40 and 5th term.If the 15th term is 127 what is its common difference?what is its first term?what is the sum of the first 30 terms.
a+9d = 40 + a+4d
a+14d = 127
solve for a and d, then find
S30 = 30/2 (2a+29d)
Let's solve the problem step by step.
Step 1: Find the common difference (d).
We know that the 10th term is equal to the sum of 40 and the 5th term.
So, T10 = T5 + 40.
Let's use the formula for the nth term of an arithmetic sequence: Tn = a + (n-1)d, where Tn is the nth term, a is the first term, and d is the common difference.
So, T10 = a + (10-1)d.
T5 = a + (5-1)d.
On substituting these values into the initial equation, we get:
a + 9d = a + 4d + 40.
Subtracting "a" from both sides, we get:
9d = 4d + 40.
Next, subtracting 4d from both sides, we have:
5d = 40.
Divide both sides of the equation by 5:
d = 8.
Therefore, the common difference is 8.
Step 2: Find the first term (a).
We are given that the 15th term (T15) is equal to 127.
Using the formula for the nth term, we have:
T15 = a + (15-1)d.
Substituting the values, we get:
127 = a + 14(8).
Simplifying, we have:
127 = a + 112.
Subtracting 112 from both sides, we get:
a = 127 - 112.
Therefore, the first term (a) is 15.
Step 3: Find the sum of the first 30 terms.
Using the formula for the sum of an arithmetic series, S = (n/2)(2a + (n-1)d), where S is the sum, n is the number of terms, a is the first term, and d is the common difference.
We want to find the sum of the first 30 terms, so n = 30.
Using the values we found earlier, a = 15 and d = 8, we can calculate the sum:
S = (30/2)(2(15) + (30-1)(8)).
S = 15(30 + 29(8)).
S = 15(30 + 232).
S = 15(262).
S = 3930.
Therefore, the sum of the first 30 terms is 3930.
To find the common difference and the first term of an arithmetic sequence, as well as the sum of the first 30 terms, we need to use the given information.
Let's break down the given information step by step:
1. The 10th term of the arithmetic sequence is equal to the sum of 40 and the 5th term.
We can write this as:
a + 9d = a + 4d + 40
Simplifying, we get:
5d = 40
Dividing both sides by 5, we find that the common difference (d) is 8.
2. The 15th term of the arithmetic sequence is 127.
We know that the general formula for arithmetic sequences is:
an = a + (n-1)d
Substituting the given values, we have:
127 = a + 14d
3. Determining the first term (a):
To find the first term, we need to solve for 'a' using the equation from step 2. Since we already know the common difference (d) is 8, we can substitute it into the equation:
127 = a + 14(8)
Simplifying further:
127 = a + 112
Subtracting 112 from both sides, we get:
a = 15
Therefore, the first term (a) is 15.
4. Finding the sum of the first 30 terms:
The sum of the first 30 terms can be found using the formula for the sum of an arithmetic series:
Sn = (n/2) * (2a + (n-1)d)
Substituting the known values, we have:
S30 = (30/2) * (2(15) + (30-1)(8))
Simplifying:
S30 = 15 * (30 + 29 * 8)
S30 = 15 * (30 + 232)
S30 = 15 * (262)
S30 = 3,930
Therefore, the sum of the first 30 terms is 3,930.