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Sequence that is the sum of two progressions with c2 and c3 given

This type of question, asked 2 times: 2021 Paper 19, 29 Jan 2023. All 2 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 19

    84
    Let a1,a2,…,a10a_{1},a_{2},\ldots,a_{10} be an AP with common difference -3 and b1,b2,….,b10b_{1},b_{2},\ldots.,b_{10} be a GP with common ratio 2 . Let ck=ak+bk,k=1,2,…,10c_{k}=a_{k}+b_{k},k= 1,2,\ldots,10. If c2=12c_{2}= 12 and c3=13c_{3}= 13, then ∑k=110ck\sum_{k = 1}^{10} c_{k} is equal to

  2. Question 2

    29 Jan 2023

    180
    Let {ak}\left\{ a_{k} \right\} and {bk},k∈N\left\{ b_{k} \right\},k \in N, be two G.P.s with common ratios r1r_{1} and r2r_{2} respectively such that a1= b1=4a_{1}= \ b_{1}= 4 and r1<r2r_{1}<r_{2}. Let ck=ak+bk,k∈Nc_{k}=a_{k}+b_{k},k \in N. If c2=5c_{2}= 5 and c3=134c_{3}=\frac{13}{4} then ∑k=1∞ck−(12a6+8 b4)\sum_{k = 1}^{\infty} c_{k}-\left( 12a_{6}+ 8{\text{ }b}_{4} \right) is equal to