The six term of a g.p -64/9 and it's common ratio is -2/3 find first term of g.p
term(6) = ar^5 = -64/9, but r = -2/3
a(-2/3)^5 = -64/9
a(-32/243) = -4/9
a = (-64/9)(-243/32)
a = 54
To find the first term of a geometric progression (g.p.), we can use the formula:
๐ = ๐กโ/๐^(๐-1)
where:
๐ is the first term
๐กโ is the sixth term
๐ is the common ratio
๐ is the position of the term we want to find
In this case, we have:
๐กโ = -64/9
๐ = -2/3
๐ = 6
Plugging in the values into the formula, we get:
๐ = (-64/9) / (-2/3)^(6-1)
= (-64/9) / (-2/3)^5
= (-64/9) / (-32/243)
= (-64/9) * (-243/32)
= 512/3
Therefore, the first term of the geometric progression is 512/3.
To find the first term of a geometric progression (g.p.), we can use the formula:
nth term = a * r^(n-1)
where:
- nth term is the term in the g.p. we want to find
- a is the first term of the g.p.
- r is the common ratio of the g.p.
- n is the position of the term in the g.p.
In this case, we are given the sixth term and the common ratio, and we need to find the first term.
Let's substitute the given values into the formula and solve for a:
-64/9 = a * (-2/3)^(6-1)
First, let's simplify the common ratio term (-2/3)^(6-1):
(-2/3)^(6-1) = (-2/3)^5 = (-2)^5 / (3)^5 = -32 / 243
Now, we can substitute this value back into the equation:
-64/9 = a * (-32 / 243)
To isolate a, let's multiply both sides of the equation by 243 to get rid of the fraction:
-64/9 * 243 = a * (-32 / 243) * 243
-64 * 27 = a * (-32)
-1728 = -32a
Dividing both sides by -32:
-1728 / -32 = a
a = 54
Therefore, the first term of the geometric progression (g.p.) is 54.