To investigate the sequence 4, 17, x, y, 56, 69, we first look for patterns or rules that govern the terms.
Starting with the first two terms:
- The first term (a_1) is 4.
- The second term (a_2) is 17.
Now, we find the difference between these two terms:
\[ a_2 - a_1 = 17 - 4 = 13 \]
Next, we investigate the difference between the second term (a_2) and the third term (a_3):
Let's denote the third term as \( x \).
If we assume the sequence grows in a linear manner, we might have a common difference that increases by a certain amount each time. Look at the next known term:
- The fourth term (a_4) is denoted as \( y \).
- The fifth term (a_5) is 56.
- The sixth term (a_6) is 69.
Now we calculate the difference between consecutive known terms \( a_5 \) and \( a_6 \):
\[ a_6 - a_5 = 69 - 56 = 13 \]
Thus, the common difference from the fifth term to the sixth term is also 13.
Next, we can create a pattern of differences:
- From a_1 to a_2, the difference is +13.
- From a_2 to a_3, if we assume a common difference of +13 works here, we can establish that:
\[ x = a_2 + 13 = 17 + 13 = 30 \]
Next, we can determine \( y \):
If we similarly assume the common difference continues, then:
\[ y = a_3 + 13 = 30 + 13 = 43 \]
Thus, we have:
- \( a_3 = 30 \)
- \( a_4 = 43 \)
Verifying the pattern with the next terms:
\[ a_5 = a_4 + 13 = 43 + 13 = 56 \] \[ a_6 = a_5 + 13 = 56 + 13 = 69 \]
Now we have the full sequence:
- \( a_1 = 4 \)
- \( a_2 = 17 \)
- \( a_3 = 30 \)
- \( a_4 = 43 \)
- \( a_5 = 56 \)
- \( a_6 = 69 \)
Now, we calculate the sum of the first 6 terms:
\[ \text{Sum} = 4 + 17 + 30 + 43 + 56 + 69 \]
Calculating step-by-step:
\[ 4 + 17 = 21 \] \[ 21 + 30 = 51 \] \[ 51 + 43 = 94 \] \[ 94 + 56 = 150 \] \[ 150 + 69 = 219 \]
So, the sum of the first 6 terms of the sequence is:
\[ \boxed{219} \]
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