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Math  /  Algebra

Question4. If the sum is 220 and the first term is 10 , find the common difference if the last term is 30. A. 2 B. 5 C. 3 D. 2/32 / 3

Studdy Solution

STEP 1

1. We are dealing with an arithmetic sequence.
2. The sum of the sequence is 220.
3. The first term of the sequence is 10.
4. The last term of the sequence is 30.
5. We need to find the common difference of the sequence.

STEP 2

1. Understand the formula for the sum of an arithmetic sequence.
2. Set up the equation using the given information.
3. Solve the equation to find the common difference.

STEP 3

Understand the formula for the sum of an arithmetic sequence. The formula is:
Sn=n2×(a+l) S_n = \frac{n}{2} \times (a + l)
where Sn S_n is the sum of the sequence, n n is the number of terms, a a is the first term, and l l is the last term.

STEP 4

Set up the equation using the given information. We know:
- Sn=220 S_n = 220 - a=10 a = 10 - l=30 l = 30
Substitute these values into the sum formula:
220=n2×(10+30) 220 = \frac{n}{2} \times (10 + 30)
Simplify the equation:
220=n2×40 220 = \frac{n}{2} \times 40

STEP 5

Solve the equation to find the number of terms n n .
220=20n 220 = 20n
Divide both sides by 20 to solve for n n :
n=22020 n = \frac{220}{20} n=11 n = 11

STEP 6

Now, use the formula for the n n -th term of an arithmetic sequence to find the common difference d d :
The n n -th term formula is:
l=a+(n−1)d l = a + (n-1)d
Substitute the known values:
30=10+(11−1)d 30 = 10 + (11-1)d
Simplify and solve for d d :
30=10+10d 30 = 10 + 10d 30−10=10d 30 - 10 = 10d 20=10d 20 = 10d d=2010 d = \frac{20}{10} d=2 d = 2
The common difference is:
2 \boxed{2}

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