3(0.15)4+3(0.15)5+3(0.15)6+⋯+3(0.15)11
If your question mean:
3 ∙ (0.15) ∙ 4 + 3 ∙ ( 0.15) ∙ 5 +3 ∙ (0.15) ∙6 +⋯+ 3 ∙ (0.15) ∙ 11
then
3 ∙ (0.15) ∙ 4 + 3 ∙ ( 0.15) ∙ 5 +3 ∙ (0.15) ∙6 +⋯+ 3 ∙ (0.15) ∙ 11 =
3 ∙ 0.15 ∙ ( 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 ) =
0.45 ∙ ( 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 )
4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 are members of arithmetic progression with initial member a1 = 4 and the common difference d = 1
You have 8 members in this progression.
Sum of the n members in A.P:
S = n ( a1 + an ) / 2
In this case:
n = 8 , a1 = 4 , an = 11
S = 8 ( 4 + 11 ) / 2 = 8 ∙ 15 / 2 = 120 / 2 = 60
3 ∙ (0.15) ∙ 4 + 3 ∙ ( 0.15) ∙ 5 +3 ∙ (0.15) ∙6 +⋯+ 3 ∙ (0.15) ∙ 11 =
0.45 ∙ ( 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 ) =
0.45 ∙ 60 = 27
I assume you meant
3(0.15)^4+3(0.15)^5+3(0.15)^6+⋯+3(0.15)^11
Given a GP with
a =3
r = 0.15
You have terms 5-12 of the geometric series. That is just
S12 - S4
Since Sn = a(1-r^n)/(1-r), that gives you
= 3(1 - 0.15^12)/(1 - 0.15) - 3(1 - 0.15^4)/(1 - 0.15) = 0.00178676
To find the value of the expression 3(0.15)4 + 3(0.15)5 + 3(0.15)6 + ⋯ + 3(0.15)11, we need to understand the pattern and then calculate each term individually.
The expression can be rewritten as:
3 * (0.15^4) + 3 * (0.15^5) + 3 * (0.15^6) + ⋯ + 3 * (0.15^11)
The pattern here is that we are multiplying each term by 0.15 raised to a progressively increasing power.
To calculate each term, we start with the base value of 0.15 and raise it to the respective power.
Let's calculate the first few terms:
Term 1: 3 * (0.15^4) = 3 * (0.001215) = 0.003645
Term 2: 3 * (0.15^5) = 3 * (0.00018225) = 0.00054675
Term 3: 3 * (0.15^6) = 3 * (0.0000273375) = 0.0000820125
You can continue this pattern and calculate the remaining terms in a similar way.
Finally, to find the sum of all these terms, add up all the individual term values.
For example, if we calculate and sum up all the terms up to Term 3:
0.003645 + 0.00054675 + 0.0000820125 = 0.0042737625
Remember to adjust the number of terms based on the scope of your calculation. In this case, the expression goes up to the 11th power, so make sure to calculate all terms up to the 11th power and sum them up to get the final result.