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Step 1: Understand the given sequences
We are given two sequences:
1) An arithmetic progression (AP) ak with common difference d = −3.
2) A geometric progression (GP) bk with common ratio r = 2.
Step 2: Define ck
We define:
ck = ak + bk, for k = 1, 2, …, 10.
It is given that c2 = 12 and c3 = 13.
Step 3: Use c2 and c3 to find a2 and b2
From c2 = 12, we have:
a2 + b2 = 12. — (1)
From c3 = 13, we have:
a3 + b3 = 13. — (2)
Step 4: Express a3 and b3 in terms of a2 and b2
Since ak is an AP with common difference d = −3:
a3 = a2 + d = a2 − 3.
Since bk is a GP with common ratio r = 2:
b3 = 2 × b2.
Step 5: Use c3 = a3 + b3 = 13 to find b2
Substituting these into c3:
a3 + b3 = (a2 − 3) + (2b2) = 13.
Also from c2 = a2 + b2 = 12, we can subtract these two equations:
(a2 − 3 + 2b2) − (a2 + b2) = 13 − 12 = 1.
Simplifying:
−3 + b2 = 1
b2 = 4.
Step 6: Find a2 using b2
From (1) a2 + b2 = 12, substituting b2 = 4:
a2 + 4 = 12
a2 = 8.
Step 7: Determine a1 and the AP terms
Since a2 = 8 and the common difference d = −3:
a1 = a2 + 3 = 8 + 3 = 11 (because going backward by the difference adds 3).
Hence the AP is 11, 8, 5, … up to a10.
Step 8: Determine b1 and the GP terms
We have b2 = 4 and common ratio r = 2.
So b1 = b2 / 2 = 4 / 2 = 2.
Hence the GP is 2, 4, 8, … up to b10.
Step 9: Find the sum of ak for k = 1 to 10
Sum of an AP with first term a1, common difference d, and n terms is given by:
S_{\text{AP}} = \frac{n}{2} [2a_{1} + (n - 1)d] .
Here, a1 = 11, d = −3, and n = 10:
S_{\text{AP}} = \frac{10}{2} [2 \times 11 + (10 - 1)(-3)] \\
= 5 \,[22 + 9 \times (-3)] \\
= 5 \, [22 - 27] \\
= 5 \, (-5) \\
= -25 .
Step 10: Find the sum of bk for k = 1 to 10
Sum of a GP with first term b1, common ratio r, and n terms is given by:
S_{\text{GP}} = b_{1} \times \frac{r^{n} - 1}{r - 1} .
Here, b1 = 2, r = 2, n = 10:
S_{\text{GP}} = 2 \times \frac{2^{10} - 1}{2 - 1} \\
= 2 \times (1024 - 1) \\
= 2 \times 1023 \\
= 2046 .
Step 11: Compute the sum of ck for k = 1 to 10
Since ck = ak + bk:
\sum_{k=1}^{10} c_{k} = \sum_{k=1}^{10} a_{k} + \sum_{k=1}^{10} b_{k} \\
= (-25) + 2046 \\
= 2021.
Final Answer:
\displaystyle 2021
