Verified answer

a1 = 81

for an = 24, r = 2/3, n = 4

an = a1(r)^(n-1) [given formula]

substitute in our known variables

24 = a1 (2/3)³

divide both sides by (2/3)³

24 / (2/3)³ = a1

reverse the two sides of the equality

a1 = 24/ (2/3)³

convert the fraction

a1 = 24 / (8/27)

multiply the numerator and the denominator on the right by 27

a1= 24(27)/8

do the arithmetic

a1 = 81

[Cheater method if you forgot the formula: You could also get this by working backward from a4 = 24 (an = 24 where n=4).

a3 = (3/2) a4 because a4 = (2/3) a3

[that's what r = 2/3 means: an+1 = (2/3)an]

<note: an+1 means 'a' with the subscript n+1, not 1 more than an>

a3 = (3/2)24

a3 = 36

a2 = (3/2) a3 also because a3 = (2/3) a3

a2 = (3/2) 36

a2 = 54

a1 = (3/2) a2

a1 = (3/2) 54

a1 = 81

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a1 = 2

for Sn = 1022, r = 2, n = 9

Sn = a1(1-r^n) / (1-r) [given formula]

substitute givens for variables

1022 = a1(1-2^9)/1-2

expand 2^9 = 2^4*2^4*2 = 16*16*2 = 256*2 = 512

1022 = a1 (1-512) / 1-2

simplify the numerator and denominator

1022 = a1 (-511) / -1

minus signs disappear an dthe denominator of 1 disappears

1022 = a1 (511)

divide both sides by 511

1022 / 511 = a1

simplify fraction

2 = a1

reverse

a1 = 2

check: series begins with a1 = 2, it increases by a factor of 2 (thats what r = 2 means) for nine factors:

2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 =

6 + 24 + 96 + 384 + 512 =

30 + 480 + 512 =

510 + 512 =

1022