The 15th term of an arithmetic series is 52, and the sum of the first 15 terms is 405 how do you find the sum of the first 22 terms?

1 Answer
Mar 27, 2016

S_22=869

Explanation:

Recall the following:

1. color(blue)(|bar(ul(color(white)(a/a)"Arithmetic sequence": t_n=a+(n-1)dcolor(white)(a/a)|)))

where:
t_n=term number
a=first term
n=number of terms
d=common difference

2. color(purple)(|bar(ul(color(white)(a/a)"Arithmetic series": S_n=n/2(2a+(n-1)d)color(white)(a/a)|)))

where:
S_n=sum of n numbers, starting at a
n=number of terms
a=first term
d= common difference

Determining the First Term and Common Difference
1. Start by using the arithmetic sequence formula to express the 15^("th") term mathematically. Label the simplified equation as equation 1.

t_n=a+(n-1)d

52=a+(15-1)d

color(darkorange)("Equation"color(white)(i)1): 52=a+14d

2. Use the arithmetic series formula to express the sum of the first 15 terms. Label the simplified equation as equation 2.

S_n=n/2(2a+(n-1)d)

405=15/2(2a+(15-1)d)

405=15/2(2a+14d)

color(darkorange)("Equation"color(white)(i)2): 405=15a+105d

3. Use elimination to subtract equation 2 from equation 1. Start by eliminating the terms with the variable, a. However, since equation 1 does not have the same coefficient for a as equation 2, multiply equation 1 by color(teal)15.

color(teal)15(52)=color(teal)15(a+14d)

780=15a+210d

4. Now that equation 1 has the same coefficient for a as equation 2, subtract equation 2 from equation 1.

color(white)(xxx)780=15a+210d
(-(405)=-(15a+105d))/(color(brown)(375=0a+105d))

5. Solve for d in color(brown)(375=0a+105d).

375=0a+105d

375=105d

color(green)(|bar(ul(color(white)(a/a)d=25/7color(white)(a/a)|)))

6. Substitute d=25/7 into either equation 1 or 2 to determine the value of a. In this case, we'll use equation 1.

780=15a+210d

780=15a+210(25/7)

780=15a+750

30=15a

color(green)(|bar(ul(color(white)(a/a)a=2color(white)(a/a)|)))

Determining the Sum of the First 22 Terms
1. Since you now have the values of a and d, you can use the arithmetic series formula to find the sum of the first 22 terms. Thus, substitute your known values into the formula.

S_n=n/2(2a+(n-1)d)

S_22=22/2(2(2)+(22-1)(25/7))

2. Solve for S_22.

S_22=11(4+21(25/7))

S_22=11(4+75)

color(green)(|bar(ul(color(white)(a/a)S_22=869color(white)(a/a)|)))