A large asteroid crashed into a moon of another planet, causing several boulders from the moon to be propelled into space toward the planet. Astronomers were able to measure the speed of one of the projectiles. The distance (in feet) that the projectile traveled each second, starting with the first second, was given by the arithmetic sequence 26, 44, 62, 80, .... Find the total distance that the projectile traveled in seven seconds.
The common difference between each term in the arithmetic sequence is 18 (44 - 26 = 18, 62 - 44 = 18, 80 - 62 = 18).
To find the total distance traveled in seven seconds, we first need to find the distance traveled in the 7th second. The 7th term in the sequence can be found using the formula for the nth term of an arithmetic sequence:
a(n) = a(1) + (n-1)d
where:
a(1) = 26 (the first term)
d = 18 (the common difference)
n = 7
a(7) = 26 + (7-1) * 18
a(7) = 26 + 6 * 18
a(7) = 26 + 108
a(7) = 134
So, in the 7th second, the projectile traveled 134 feet.
To find the total distance traveled in seven seconds, we need to sum up the distances traveled in each second:
26 + 44 + 62 + 80 + 98 + 116 + 134 = 560
Therefore, the total distance the projectile traveled in seven seconds is 560 feet.