Certainly! Let's carefully solve this step by step.
We are given:
- The first term of the sequence ([tex]\(a_1\)[/tex]) is 12.
- The first term of the first level difference ([tex]\(d_1\)[/tex]) is 5.
- The constant difference ([tex]\(c\)[/tex]) is 6.
- We need to find the next 6 terms of the sequence.
Let's denote:
- The sequence as [tex]\(a_n\)[/tex], where [tex]\(n\)[/tex] is the term number.
- The first level differences as [tex]\(D_n\)[/tex].
The first level difference between terms is defined by [tex]\(D_n = a_{n+1} - a_n\)[/tex].
Given data:
- [tex]\(a_1 = 12\)[/tex]
- [tex]\(D_1 = 5\)[/tex]
- Constant difference [tex]\(c = 6\)[/tex]
The first level differences [tex]\(D_n\)[/tex] are formed by adding a constant difference [tex]\(c\)[/tex] to the previous first level difference:
[tex]\[ D_{n+1} = D_n + c \][/tex]
So we need to write the terms of the sequence, and we start with [tex]\(a_1 = 12\)[/tex]:
### Step-by-Step Calculation:
1. First level differences [tex]\(D_n\)[/tex] Calculation:
- [tex]\( D_1 = 5 \)[/tex]
- [tex]\( D_2 = D_1 + c = 5 + 6 = 11 \)[/tex]
- [tex]\( D_3 = D_2 + c = 11 + 6 = 17 \)[/tex]
- [tex]\( D_4 = D_3 + c = 17 + 6 = 23 \)[/tex]
- [tex]\( D_5 = D_4 + c = 23 + 6 = 29 \)[/tex]
- [tex]\( D_6 = D_5 + c = 29 + 6 = 35 \)[/tex]
2. Sequence Calculation [tex]\(a_n\)[/tex] :
- [tex]\( a_1 = 12 \)[/tex]
- [tex]\( a_2 = a_1 + D_1 = 12 + 5 = 17 \)[/tex]
- [tex]\( a_3 = a_2 + D_2 = 17 + 11 = 28 \)[/tex]
- [tex]\( a_4 = a_3 + D_3 = 28 + 17 = 45 \)[/tex]
- [tex]\( a_5 = a_4 + D_4 = 45 + 23 = 68 \)[/tex]
- [tex]\( a_6 = a_5 + D_5 = 68 + 29 = 97 \)[/tex]
- [tex]\( a_7 = a_6 + D_6 = 97 + 35 = 132 \)[/tex]
### Solution:
The next 6 terms of the sequence are:
- [tex]\( a_2 = 17 \)[/tex]
- [tex]\( a_3 = 28 \)[/tex]
- [tex]\( a_4 = 45 \)[/tex]
- [tex]\( a_5 = 68 \)[/tex]
- [tex]\( a_6 = 97 \)[/tex]
- [tex]\( a_7 = 132 \)[/tex]
So, the sequence after the first term is:
[tex]\[ 17, 28, 45, 68, 97, 132 \][/tex]
