Let's analyze the sequence and find how many terms will give a sum of 9.
Step 1: Identify the sequence pattern
The sequence given is: 1, 3, 7, … Let's check the pattern between terms:
- From 1 to 3: increase by 2
- From 3 to 7: increase by 4
The differences are increasing: 2, 4, ... Let's check if the sequence is defined by a formula. Check if it's a geometric or arithmetic sequence:
- Not arithmetic because differences are not constant.
- Not geometric because ratios are not constant (3/1=3, 7/3≈2.33).
Try to find a pattern for the nth term. Look at the terms:
- a1=1a_1=1a1=1
- a2=3a_2=3a2=3
- a3=7a_3=7a3=7
Try to write ana_nan: Notice that a1=21−1=1a_1=2^1-1=1a1=21−1=1 a2=22−1=4−1=3a_2=2^2-1=4-1=3a2=22−1=4−1=3 a3=23−1=8−1=7a_3=2^3-1=8-1=7a3=23−1=8−1=7 So the nth term is:
an=2n−1a_n=2^n-1an=2n−1
Step 2: Find the sum of the first n terms
Sum of first n terms:
Sn=∑k=1nak=∑k=1n(2k−1)=∑k=1n2k−∑k=1n1=(∑k=1n2k)−nS_n=\sum_{k=1}^na_k=\sum_{k=1}^n(2^k-1)=\sum_{k=1}^n2^k-\sum_{k=1}^n1=\left(\sum_{k=1}^n2^k\right)-nSn=k=1∑nak=k=1∑n(2k−1)=k=1∑n2k−k=1∑n1=(k=1∑n2k)−n
Sum of powers of 2 from 1 to n:
∑k=1n2k=2n+1−2\sum_{k=1}^n2^k=2^{n+1}-2k=1∑n2k=2n+1−2
Therefore,
Sn=(2n+1−2)−n=2n+1−2−nS_n=(2^{n+1}-2)-n=2^{n+1}-2-nSn=(2n+1−2)−n=2n+1−2−n
Step 3: Find n such that Sn=9S_n=9Sn=9
Set the sum equal to 9:
2n+1−2−n=92^{n+1}-2-n=92n+1−2−n=9
Rearranged:
2n+1−n=112^{n+1}-n=112n+1−n=11
Try integer values of n:
- For n=1n=1n=1: 22−1=4−1=3≠112^{2}-1=4-1=3\neq 1122−1=4−1=3=11
- For n=2n=2n=2: 23−2=8−2=6≠112^{3}-2=8-2=6\neq 1123−2=8−2=6=11
- For n=3n=3n=3: 24−3=16−3=13≠112^{4}-3=16-3=13\neq 1124−3=16−3=13=11
- For n=4n=4n=4: 25−4=32−4=28≠112^{5}-4=32-4=28\neq 1125−4=32−4=28=11
Check n=3n=3n=3 sum explicitly:
S3=24−2−3=16−2−3=11S_3=2^{4}-2-3=16-2-3=11S3=24−2−3=16−2−3=11
Wait, this is 11, not 9. Try n=2n=2n=2:
S2=23−2−2=8−2−2=4S_2=2^{3}-2-2=8-2-2=4S2=23−2−2=8−2−2=4
Try n=1n=1n=1:
S1=22−2−1=4−2−1=1S_1=2^{2}-2-1=4-2-1=1S1=22−2−1=4−2−1=1
Try n=0n=0n=0:
S0=0S_0=0S0=0
Try n=3n=3n=3 sum is 11, so no n gives exactly 9. Try n=2.5n=2.5n=2.5 (not integer, but just to check): Between n=2n=2n=2 and n=3n=3n=3, sum jumps from 4 to 11, so no integer nnn gives sum 9.
Step 4: Check sum of first few terms manually
- Sum of first term: 1
- Sum of first two terms: 1 + 3 = 4
- Sum of first three terms: 1 + 3 + 7 = 11
No sum equals 9.
Conclusion:
No integer number of terms in the sequence 1, 3, 7, ... (where an=2n−1a_n=2^n-1an=2n−1) will give a sum exactly equal to 9. If you want, I can help you find the closest sum to 9 or explore other sequences!
