Question

use the graphs of the arithmetic sequences \\{a_n\\} and \\{b_n\\} to find a_{17} + b_{10}.

Explanation:

Step1: Analyze sequence \(\{a_n\}\)

For arithmetic sequence \(\{a_n\}\), let's find the first term \(a_1\) and common difference \(d_a\). From the graph, when \(n = 1\), \(a_1=-10\); when \(n = 2\), \(a_2=-6\). So \(d_a=a_2 - a_1=-6-(-10) = 4\). The formula for arithmetic sequence is \(a_n=a_1+(n - 1)d_a\). So \(a_{17}=a_1+(17 - 1)d_a=-10+16\times4=-10 + 64 = 54\)? Wait, no, wait the graph of \(\{a_n\}\): Wait, maybe I misread the graph. Wait the first graph (for \(\{a_n\}\)): let's check the coordinates. Let's assume the x - axis is \(n\) (term number) and y - axis is \(a_n\). Wait, maybe the first point (n = 1) is at (1, - 10), n = 2 at (2, - 6), n = 3 at (3, - 2), n = 4 at (4, 2), n = 5 at (5, 6)? Wait no, the y - axis for \(\{a_n\}\) has labels - 12, - 10, - 8, - 6, - 4, - 2, 0, 2, 4. Wait, maybe the first point (n = 1) is (1, - 10), n = 2 (2, - 6), n = 3 (3, - 2), n = 4 (4, 2), n = 5 (5, 6). So common difference \(d_a=4\). Then \(a_n=-10+(n - 1)\times4=4n-14\). So \(a_{17}=4\times17-14=68 - 14 = 54\)? Wait no, wait 4*17=68, 68 - 14 = 54. Now for \(\{b_n\}\): the second graph, x - axis \(n\), y - axis \(b_n\). Let's find \(b_1\) and \(d_b\). Let's say the first point (n = 1) is (1, 1), n = 2 (2, 3), n = 3 (3, 5), n = 4 (4, 7), n = 5 (5, 9)? Wait no, the y - axis for \(\{b_n\}\) has labels 0, 2, 4, 6. Wait, maybe the first point (n = 1) is (1, 1), n = 2 (2, 3), n = 3 (3, 5), n = 4 (4, 7), n = 5 (5, 9). Wait, no, let's check the graph again. Wait the second graph: the first point (n = 1) is at (1, 1) (y - axis 1), n = 2 at (2, 3), n = 3 at (3, 5), n = 4 at (4, 7), n = 5 at (5, 9). So common difference \(d_b = 2\). The formula for \(b_n=b_1+(n - 1)d_b\). If \(b_1 = 1\), then \(b_n=1+(n - 1)\times2=2n - 1\). Wait, but let's check \(b_{10}\): \(b_{10}=2\times10-1 = 19\)? Wait no, maybe I misread the graph. Wait, maybe the first point of \(\{b_n\}\) is (1, 1), n = 2 (2, 3), n = 3 (3, 5), n = 4 (4, 7), n = 5 (5, 9). So \(b_n=2n - 1\). Then \(b_{10}=2\times10-1 = 19\). But wait, maybe the first term of \(\{a_n\}\) is different. Wait, let's re - examine \(\{a_n\}\) graph. Let's take n = 1: y = - 10, n = 2: y = - 6, n = 3: y = - 2, n = 4: y = 2, n = 5: y = 6. So \(a_n=-10+(n - 1)\times4=4n-14\). So \(a_{17}=4\times17-14=68 - 14 = 54\). Now for \(\{b_n\}\), let's check the graph again. The second graph: the points are at (1, 1), (2, 3), (3, 5), (4, 7), (5, 9). So it's an arithmetic sequence with \(b_1 = 1\), \(d_b=2\). So \(b_n=1+(n - 1)\times2=2n - 1\). Then \(b_{10}=2\times10-1 = 19\). Then \(a_{17}+b_{10}=54 + 19=73\)? Wait, no, maybe I made a mistake in the graph reading. Wait, maybe the \(\{a_n\}\) graph: let's check the y - axis. The first graph's y - axis is labeled from - 12 to 4. Let's take n = 1: (1, - 10), n = 2: (2, - 6), n = 3: (3, - 2), n = 4: (4, 2), n = 5: (5, 6). So the common difference is 4. So \(a_n=-10+(n - 1)\times4=4n-14\). Then \(a_{17}=4*17 - 14=68 - 14 = 54\). For \(\{b_n\}\), the second graph: x - axis n, y - axis b_n. Let's see the points: (1, 1), (2, 3), (3, 5), (4, 7), (5, 9). So \(b_n=2n - 1\). Then \(b_{10}=2*10 - 1 = 19\). Then \(a_{17}+b_{10}=54 + 19 = 73\). Wait, but maybe the \(\{a_n\}\) has a different first term. Wait, maybe n = 1 is (1, - 10), n = 2 (2, - 6), so \(d_a = 4\), \(a_1=-10\). Then \(a_n=a_1+(n - 1)d_a=-10 + 4(n - 1)=4n-14\). Correct. For \(\{b_n\}\), let's check n = 1: y = 1, n = 2: y = 3, so \(d_b = 2\), \(b_1 = 1\), so \(b_n=1+2(n - 1)=2n - 1\). So \(b_{10}=2*10 - 1 = 19\). Then \(a_{17}+b_{10}=4*17-14+2*10 - 1=68 - 14+20 - 1=54 + 19 = 73\).

Step2: Re - check the arithmetic sequence formulas

For \(\{a_n\}\):
We have \(a_1=-10\), \(d_a = 4\). The formula for the \(n\)th term of an arithmetic sequence is \(a_n=a_1+(n - 1)d\). So \(a_n=-10+(n - 1)\times4=4n-14\).
For \(n = 17\):
\(a_{17}=4\times17-14=68 - 14 = 54\).
For \(\{b_n\}\):
We have \(b_1 = 1\), \(d_b = 2\). The formula for the \(n\)th term of an arithmetic sequence is \(b_n=b_1+(n - 1)d\). So \(b_n=1+(n - 1)\times2=2n - 1\).
For \(n = 10\):
\(b_{10}=2\times10-1=20 - 1 = 19\).

Step3: Calculate \(a_{17}+b_{10}\)

Now, we add the two values: \(a_{17}+b_{10}=54 + 19=73\).

Answer:

73